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MATHEMATICAL  TEXTS 
FOR  SCHOOLS 

EDITED  Br 

PERCEY  F.  SMITH,  Ph.D. 

PROFESSOR  OF  MATHEMATICS   IN  THE  SHEFFIELD 
SCIENTIFIC   SCHOOL   OF  YALE  UNIVERSITY 


FIRST  COURSE  IN  ALGEBRA 


BY 
HERBERT  E.  HAWKES,  Ph.D. 

PROFESSOR  OF  MATHEMATICS  IN  COLUMBIA   UNIVERSITY 


WILLIAM  A.  LUBY,  A.B. 

HEAD  OF  THE  DEPARTMENT  OF  MATHEMATICS 
KANSAS  CITY  POLYTECHNIC  INSTITUTE 


FRANK  C.  TOUTOlSr,  A.M.  *       ' 

FORMERLY  PRINCIPAL  OF   CENTRAL  Hlt^H  SCEOOlT 
ST.  JOSEPH,  MISSOURI  ' 


REVISED  EDITION 


GINN  AND  COMPANY 

BOSTON     •    NEW   YORK     •CHICAGO     •     LONDON 
ATLANTA     •    DALLAS     •COLUMBUS        ^AN   FRANCISCO 


1911 


ENTERED  AT  STATIONERS'  HALL 


COPYRIGHT,  1909,  1910,  1917,  BY 

herbert  e.  hawkes,  william  a.  lubt 
'  ":  • /';  ;  and  frank  c.  touton 


/^    i£t  RIGHTS  RESERVED 
*'•*    '  A321.7 


GINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A- 


PREFACE 

In  this  revision  of  their  "First  Course  in  Algebra''  the  authors 
have  in  general  followed  the  plan  of  that  text  in  the  order  of 
topics  treated  and  in  the  method  of  their  presentation. 

The  most  important  modification  of  the  order  of  topics  is 
found  in  the  transference  of  the  work  on  Eatio  and  Propor- 
tion to  the  last  chapter  in  the  book  and  the  omission  of  the 
chapter  on  the  Highest  Common  Factor  and  Lowest  Common 
Multiple.  The  latter  topic  is  treated  in  connection  with  the 
related  material  on  fractions,  while  the  former  is  placed  among 
the  Supplementary  Topics  at  the  end  of  the  book. 

Material  for  which  there  is  no  strong  demand  from  teachers 
has  been  omitted,  and  the  entire  work  has  been  rewritten  in 
the  interest  of  greater  simplicity  and  directness  of  appeal.  The 
collections  of  exercises  and  problems  ar©  for  the  most  part  new 
and  contain  a  larger  proportion  of  easy  exereises  with  simple 
results  than  the  first  edition. 

A  striking  feature  of  the  revision  is  the  inclusion  of  a  large 
number  of  oral  exercises  in  connection  with  the  introduction  of 
each  new  idea  or  operation.  It  is  the  object  of  these  exercises 
to  present  the  new  concept  in  complete  isolation  from  any  com- 
plication of  notation  or  technique  so  that  the  student  becomes 
familiar  with  its  content  and  bearing  before  he  is  asked  to 
make  use  of  it  in  written  work.  These  oral  exercises  may  well 
be  taken  up  when  the  advance  lesson  is  assigned,  so  that  the 
pupil  may  be  certain  that  he  understands  the  idea  involved 
in  the  new  work  before  he  leaves  his  instructor. 

Another  feature  scarcely  less  important  is  the  character  and 
position   of  the   examples  and  hints.    The  aim  has  been  to 

V 

459907 


vi  FIRST  COURSE  IN  ALGEBRA 

help  the  student  at  the  exact  point  where  he  needs  it  and 
to  avoid  the  insertion  of  lengthy  and  difficult  solutions  before 
they  can  be  completely  understood. 

The  definitions  and  axioms  have  been  expressed  in  the  sim- 
plest language  which  is  consistent  with  scientific  accuracy. 
Many  definitions  which  are  usually  found  in  elementary  texts 
but  which  do  not  contribute  to  the  clearness  of  the  subject 
are  omitted. 

The  first  presentation  of  the  subject  of  graphs  has  been  limited 
to  the  study  of  the  straight  line  and  a  few  exercises  of  a  com- 
mercial or  scientific  character.  These  exercises  not  only  have 
a  very  definite  human  interest  apart  from  their  mathematical 
value  but  also  serve  to  familiarize  the  student  with  the  kind 
of  graphs  he  will  meet  in  his  ordinary  reading. 

The  first  consideration  in  the  treatment  of  radicals  has  been 
the  needs  of  the  student  for  his  later  study  of  the  quadratic 
equation  and  for  his  work  in  geometry. 

Frequently  a  student's  knowledge  of  algebra  is  limited  to  a 
greater  or  less  facility  in  the  use  of  the  rules  of  operation  — 
to  mere  technique.  To  obviate  this  result  the  development  of 
the  problem  work  in  this  text  has  received  full  and  careful 
attention. 

The  authors  have  received  suggestions  of  great  value  from 
many  teachers  in  all  parts  of  the  country,  for  which  they 
extend  their  thanks.  They  are  under  especial  obligation  to 
Mr.  E.  L.  Brown,  of  Denver,  Colorado,  Professor  H.  E.  Cobb, 
of  Chicago,  Illinois,  and  to  Mr.  L.  A.  Pultz,  of  Rochester, 
New  York,  for  helpful  criticism. 


CONTENTS 

CHAPTER  PAGE 

I.   Introduction  (Sects.  1-11) 1 

II.   Positive  and  Negative  I^umbers  (Sects.  12-19)  .  18 

III.  Addition  (Sects.  20-25) 33  -^ 

IV.  Simple  Equations  (Sects.  26-28) 39 

V.    Subtraction  (Sects.  29-30) 49  '^ 

VL   Identities  and  Equations  of  Condition  (Sects. 

31-34) 54 

VII.   Parentheses  (Sects.  35-36) 64 

VIII.   Multiplication  (Sects.  37-44) 70-^ 

IX.   Parentheses  in  Equations  (Sects.  45-46)     ...  79 

X.   Division  (Sects.  47-49) 87  -^ 

XI.   Equations  and  Problems  (Sects.  50-51)  ....  95 

XII.   Important  Special  Products  (Sects.  52-55)      .     .  105 

XIII.  Factoring  (Sects.  56-66) 113 

XIV.  Solution    of    Equations    by    Factoring    (Sects. 

67-70) 137 

XV.   Fractions  (Sects.  71-81) 148 

XVI.    Equations  containing  Fractions  (Sects.  82-88)  .  175  ^ 

XVIL   Graphical  Representation  (Sects.  89-94)    ...  200 

XVIII.    Linear  Systems  (Sects.  95-100) 217 

XIX.   Square  Root  (Sects.  101-102) 240 

XX.   Radicals  (Sects.  103-114) 250 

XXI.    Quadratic  Equations  (Sects.  115-117)     ....  270 

XXII.   Ratio  and  Proportion  (Sects.  118-125)    ....  282 

Supplementary  Topics  (Sects.  126-130) 293 

Index 307 

vii 


ILLUSTRATIONS 


PAGE 

JOHN  WALLIS 48 

SIR  WILLIAM  ROWAN  HAMILTON 70 

SIR  ISAAC  NEWTON       100 

JOHN  NAPIER 186 

REN:^  DESCARTES 210 

FRANCOIS  VIETA 268 


IX 


FIRST  COURSE  IN  ALGEBRA 


CHAPTER  I 

INTRODUCTION 

1.  The  numbers  and  symbols  of  arithmetic.  The  simple 
operation  of  counting  employs  the  numbers  we  call  in- 
tegers. To  represent  these  integers  and  the  other  numbers 
with  which  it  deals,  arithmetic  uses  the  nuriierals  0,  1, 
2,  3,  4,  5,  6,  7,  8,  and  9.  Operations  on  the  numbers  of 
arithmetic  are  indicated  by  the  symbols  +,  — ,  X,  and  -5-. 
The  operation  of  division  applied  to  integers  gives  rise 
to  fractions.  With  these  two  kinds  of  numbers,  integers 
and  fractions,  the  student's  work  in  arithmetic  is  mainly 
carried  on. 

2.  Symbols  of  algebra.  Symbols  are  employed  far  more 
extensively  in  algebra  than  in  arithmetic,  and  many  new 
ideas  arise  in  connection  with  their  meaning  and  their  use. 
Some  symbols  represent  numbers,  others  indicate  opera- 
tions upon  them,  others  represent  relations  between  them, 
and  still  others  represent  kinds  of  numbers  with  which 
arithmetic  does  not  deal.  Letters  as  well  as  arable  numer- 
als are  used  to  represent  numbers.  The  following  symbols 
of  operation,  -f ,  — ,  X,  and  -h,  have  the  same  meaning  as  in 
arithmetic.  The  sign  of  multiplication  is  usually  replaced 
by  a  dot  or  omitted. 

For  example,  3  x  a  is  written  3  •  a,  or  3  a,  and  2  x  a  x  b  is 
written  2  ab.    Also  a  -^  &  is  often  written  - . 


2  FIRST  COUBSE  IK  ALGEBKA 

The  sign  =  is  read  equals^  or  is  equal  to.  As  the  need 
for  them  arises,  other  symbgls  will  be  introduced. 

3.  The  use  of  letters  to  represent  numbers.  The  use  of 
the  letters  of  the  alphabet  to  represent  numbers  is  the 
most  striking  difference  between  arithmetic  and  algebra. 

In  arithmetic  we  speak  thus:  If  the  sides  of  a  triangle 
are  6,  7,  and  9  inches  respectively,  its  perimeter  is  6  +  7  +  9, 
or  22  mches.  The  corresponding  statement  in  algebra  is: 
If  the  sides  of  a  triangle  are'  a,  J,  and  c  inches  respectively, 
and  its  perimeter  is  p  inches,  then  p  =  a-\-h-\-  c.  Here  the 
second  statement  is  true  for  every  triangle,  while  the  first 
is  not  true  for  every  triangle. 

Similarly:  If  a  rectangle  is  8  inches  by  12  inches,  its 
perimeter  is  8+12  +  8+12,  or  40  inches.  And  if  a  rec- 
tangle is  I  inches  long  and  w  mches  wide  and  if  p  denotes 
its  perimeter  in  inches,  then  p  =  l-{-w  -\-l-{-w^  or  2l-\-2iv. 
Here,  again,  the  arithmetical  statement  is  particular  and 
applies  to  one  rectangle  only,  while  the  algebraic  state- 
ment is  general;  that  is,  it  is  true  for  all  rectangles. 

The  gain  in  power  which  the  general  symbolic  language 
of  algebra  affords  over  the  particular  numerical  language 
of  arithmetic  constitutes  one  of  the  most  important  advan- 
tages of  the  algebraic  method.  As  the  student  progresses  he 
will  meet  with  many  illustrations  of  this  feature  of  algebra. 

The  purpose  of  the  following  exercises  is  to  familiarize 
the  student  with  the  use  of  letters  in  the  place  of  numbers. 

ORAL  EXERCISES 

1.  What  numerical  value  has  5  a  when  <z  is  3  ?  when  a.  is  5  ? 
when  a  is  10  ? 

2.  What  numerical  value  has  6a +  2^  when  a  is  2  and  ^  is  4? 

3.  Express  A  +  S?/^  in  seconds  if  h  and  m  stand. for  the 
number  of  seconds  in  an  hour  and  in  a  minute  respectively. 


INTEODUCTION  3 

4.  Express  10^  +  4/  in  inches  if  y  and  /  stand  for  the 
number  of  inches  in  a  yard  and  in  a  foot  respectively. 

5.  If  g'  and  d  represent  the  number  of  cents  in  a  quarter 
and  in  a  dime  respectively,  express  4  5'  +  6  (i  in  cents. 

6.  If  2^  and  h  represent  the  number  of  pounds  in  one  ton 
and  in  one  hundredweight  respectively,  express  4^  +  6/2,  in 
pounds. 

7.  3  cc  +  5  i«  =  how  many  cc  ?  9.  5  •  ?^  +  10  •  j^  =  (?)  2^. 

8.  4x  +  5cc=(?)x.  10.  5:^  +  3i:c  +  6a;  =  ? 

11.  Zx-2x-\-1x-^x  =  ? 

12.  7  books  +  3  chairs  +  2  books  +  5  chairs  =  (?)  books  -f- 
(?)  chairs. 

13.  8  books  +  4  chairs  —  2  chairs  +  4  books  =  (?)  books  + 
(?)  chairs. 

14.  6  books  +  7  chairs  —  3  books  —  2  chairs  =  (?)  books  + 
(?)  chairs. 

15.  55  +  4c4-8Z^-2c  =  (?)5+(?)c. 

16.  6^  +  3cc  +  4&  +  8a:  =  (?)^»  +  (?)a^. 

17.  ^x-{-2h-\-'^x-h^x  =  {^)h^(^)x, 

18.  2aj  +  2  +  3^  +  4  =  (?)a;  +  ? 

19.  4;r  +  2  +  3:r  +  2-iZJ  +  8  =  ? 

20.  ic+x  +  2  +  cc  +  ^  +  2  =  ?        22.  5a4-18-3a-7=? 

21.  714-^  +  1  +  ^  +  2=?  23.  8::c-3+18-5ic=? 

24.  4i^-8  +  3i^  +  20  =  ? 

25.  What  value  has  4  a;  +  3  when  iic  =  2  ?  when  x  =  1  ? 

26.  What  value  has  2>x  —  A:  when  cc  =  3  ?  when  x  =  2? 

27.  The  side  of  a  square  is  5  inches  long.   What  is  its  area? 
its  perimeter  ? 


4  FIRST  COURSE  IN  ALGEBRA 

28.  The  side  of  a  square  is  s  inches  long.  What  represents 
its  perimeter  ?  its  area  ? 

29.  The  base  of  a  rectangle  is  12  feet,  and  its  altitude  is 
4  feet.   What  is  its  perimeter  ?  its  area  ? 

30.  li  b  represents  the  number  of  feet  in  the  base  of  a  rec- 
tangle and  a  the  number  of  feet  in  its  altitude,  what  is  its 
perimeter  ?  its  area  ? 

31.  A  rectangle  is  twice  as  long  as  it  is  wide.  Let  w  repre- 
sent the  number  of  inches  in  its  width.  Then  express,  in  terms 
of  w,  (a)  the  length ;  (b)  the  perimeter ;  (c)  the  area. 

32.  A  man  is  three  times  as  old  as  his  son.  If  s  denotes 
the  number  of  years  in  the  son's  age,  what  will  represent  the 
father's  age  ? 

33.  A  father  is  28  years  older  than  his  son.  If  s  represents 
the  son's  age  in  years,  what  will  represent  the  father's  age  ? 

34.  A  rectangle  is  24  inches  longer  than  it  is  wide.  Let  b 
represent  the  width  in  feet.  Then  represent  the  length  and  the 
perimeter  in  terms  of  b  and  numbers. 

35.  A  rectangle  is  16  feet  narrower  than  it  is  long.  If  w 
represents  the  width  in  feet,  what  will  conveniently  represent 
the  length  ?  the  perimeter  ? 

36.  A  rectangle  is  4  feet  longer  than  twice  its  width.  Express 
the  width,  the  length,  and  the  perimeter  in  terms  of  a  letter,  or 
a  letter  and  numbers. 

Origin  of  symbols.  Many  of  the  symbols  that  are  in  common  use 
in  algebra  at  the  present  time  have  histories  which  not  only  are 
interesting  in  themselves,  but  which  also  serve  to  indicate  the  slow 
and  uncertain  development  of  the  subject.  It  is  often  found  that 
symbols  which  seem  without  meaning  represent  some  abbreviation 
or  suggestion  long  since  forgotten,  and  that  operations  and  methods 
which  we  find  hard  to  master  have  sometimes  required  hundreds  of 
years  to  perfect. 


INTEODUCTION    V  6 

In  the  early  centuries  there  were  practically  no  algebraic  symbols 
in  common  use ;  one  wrote  out  in  full  the  words  plus,  minus,  equals, 
and  the  like.  But  in  the  sixteenth  century  several  Italian  mathema- 
ticians used  the  initial  letters  p  and  m  for  +  and  — .  Some  think 
that  our  modern  symbol  —  came  into  use  through  writing  the  initial 
m  so  rapidly  that  the  curves  of  the  letter  gradually  flattened  out, 
leaving  finally  a  straight  line.  The  symbol  +  may  have  originated 
similarly  in  the  rapid  writing  of  the  letter  p.  But  in  the  opinion  of 
others  these  symbols  were  first  used  in  the  German  warehouses  of 
the  fifteenth  century  to  mark  the  weights  of  boxes  of  goods.  If  a 
lot  of  boxes,  each  supposed  to  weigh  100  pounds,  came  to  the  ware- 
house, the  weight  would  be  checked,  and  if  a  certain  box  exceeded 
the  standard  weight  by  5  pounds,  it  was  marked  100  +  5 ;  if  it 
lacked  5  pounds,  it  was  marked  100  —  5.  Though  the  first  book  to 
use  these  symbols  was  published  in  1489,  it  was  not  until  about 
1630  that  they  could  be  said  to  be  in  common  use. 

Both  of  the  symbols  for  multiplication  given  in  the  text  were 
first  used  about  1630.  The  cross  was  used  by  two  Englishmen, 
Oughtred  and  Harriot,  and  was  probably  an  adaptation  of  the  letter 
X,  which  is  found  some  years  earlier.  The  dot  is  first  found  in  the 
writings  of  the  Frenchman  Descartes.  It  is  interesting  to  note  that 
Harriot  was  sent  to  America  in  1585  by  Sir  Walter  Raleigh,  and 
returned  to  England  with  a  report  of  observations.  He  made  the 
first  survey  of  Virginia  and  North  Carolina,  and  constructed  maps 
of  those  regions. 

It  is  strange  that  the  line  was  used  to  denote  division  long  before 
any  of  the  other  symbols  here  mentioned  were  in  use.  This  is,  in 
fact,  one  of  the  oldest  signs  of  operation  that  we  have.    The  Arabs, 

as  early  as  1000  a.d.,  used  both  -  and  a/b  to  denote  the  quotient  of 

a  and  b.   The  symbol  -^  did  not  occur  until  about  1630. 

Equality  has  been  denoted  in  a  variety  of  ways.  The  word  equals 
was  usually  written  out  in  full  until  about  the  year  1600,  though 
the  two  sides  of  an  equation  were  written  one  over  the  other  by  the 
Hindus  as  early  as  the  twelfth  century.  The  modem  sign  =  was 
probably  introduced  by  the  Englishman  Recorde,  in  1557,  because, 
he  says,  "  Noe.  2.  thynges  can  be  moare  equalle  "  than  two  parallel 
lines.  This  symbol  was  not  generally  accepted  at  first,  and  in  its 
place  the  symbols  II,  x,  and  go  are  frequently  met  during  the  next 
fifty  years. 


6  WEST  COUBSE  IN  ALGEBRA 

4.  The  usefulness  of  symbols.  Symbols  enable  one  to 
abbreviate  ordinary  language  in  the  solution  of  problems. 

For  example :  Three  times  a  certain  number  is  equal 
to  20  diminished  by  5.    What  is  the  number? 

If  n  represents  the  number,  the  preceding  statement  and 
'  question  can  be  written  in  symbols,  thus : 

3  7i  =  20  -  5. 
/i  =  ? 

The  symbolic  statement  3  /i  =  20  —  5  is  called  an  equa- 
tion and  n  the  unknown  number. 

If  3/1  =  20-5, 

then  3ri=15, 

and  71  =  5. 

While  the  preceding  example  is  very  simple,  it  illus- 
trates the  algebraic  method  of  stating  and  solving  the 
problem.  The  method  is  brief  and  direct,  and  its  advan- 
tages will  become  more  apparent  as  the  student  progresses. 

ORAL  EXERCISES 

Find  the  numerical  value  of  x  in  the  following  equations : 

1.  3aj  =  18.  4.  7x  =  42.  7.  6a;  =  17 +  13 

2.  4x  =  28.  5.  4a;  =  12  +  8.  8.  4ic+3x  =  35. 

3.  5aj  =  30.  6.  3aj  =  4-f  11.         9.  6a;-}-2x  =  32. 

10.  5  X  +  4  a;  =  45.  14.  4  x  —  a?  =  15  —  6. 

11.  4a;-f  3x  =  56.  15.  5x  +  4a;  —  2ic  =  10  +  4. 

12.  7x  +  2ic  =  15  +  3.        16.  4a;  +  3a;  -  a;  =  33  -  3. 

13.  9a;-3a;  =  18 -hl2.      17.  6a;  -  a; -f  3a;  =  42  +  6. 
18.  If  one  number  is  represented  by  a;,  what  will  represent 

a  number  three  times  as  great  ? 


INTRODUCTION  7 

19.  James  had  3x  cents.  His  brother  had  four  times  as 
many.    Represent  the  number  of  cents  the  brother  had. 

20.  Paul's  weight  is  2  jr  pounds,  and  his  father  weighs  three 
times  as  much.  What  will  represent  the  father's  weight  ?  the 
weight  of  the  two  together  ? 

21.  The  area  of  a  circle  is  6y.  Represent  the  area  of  a 
circle  three  times  as  large. 

22.  One  number  is  twice  a  second,  and  the  second  is  four 
times  the  third.  If  x  represents  the  third,  what  will  represent 
the  second  ?   the  first  ? 

23.  One  newsboy  has  three  times  as  many  papers  as  a 
second,  and  the  two  together  have  as  many  as  a  third. 
Represent  in  terms  of  x  the  number  of  papers  each  has. 

EXAMPLE 

The  sum  of  two  numbers  is  112.  The  greater  is  three  times 
the  less.   What  are  the  numbers  ? 

Solution.    By  the  conditions  of  the  problem, 

greater  number  +  less  number  =  112.  (1) 

Let  I  =  the  less  number. 

Then  3  Z  =  the  greater  number. 

Substituting  these  symbols  in  (1),  we  have 
31  +  1  =  112. 

Collecting,  4  I  =  112.- 

Whence  Z  =  i  ji  =  28, 

and  3  /  =  3  X  28  =  84. 

Therefore  the  greater  number  is  84  and  the  less  28. 

We  may  verify  the  result  by  substituting  84  and  28  in  the  problem. 

Thus  84  +  28  =  112, 

and  84  =  3  .  28. 


8  FIEST  COURSE  IN  ALGEBEA 

PROBLEMS 

1.  The  sum  of  two  numbers  is  120.  The  greater  is  four 
times  the  less.    Find  each. 

2.  A  certain  number  plus  seven  times  itself  equals  216. 
Find  the  number. 

3.  One  number  is  eight  times  another.  Their  sum  is  72. 
Find  each. 

4.  The  first  of  three  numbers  is  twice  the  third,  and  the 
second  is  four  times  the  third.  The  sum  of  the  three  numbers 
is  252.    Find  the  numbers. 

Hint.  Let  x  =  the  third  number.  Then  2  x  =  the  first,  and  4  x  =  the 
Second. 

5.  The  sum  of  three  numbers  is  117.  The  second  is  twice 
the  first,  and  the  third  is  three  times  the  second.    Find  each. 

6.  There  are  three  numbers  whose  sum  is  192.  The  first 
is  twice  the  second,  and  the  third  equals  the  sum  of  the  other 
two.    Find  the  numbers. 

7.  The  sum  of  three  numbers  is  312.  The  second  is  five 
times  the  first,  and  the  third  is  four  times  the  second.  Find 
the  numbers. 

8.  The  sum  of  three  numbers  is  208.  The  second  is  three 
times  the  first,  and  the  third  is  the  sum  of  the  other  two. 
Find  the  numbers. 

9.  A  man  is  three  times  as  old  as  his  son.  The  sum  of 
their  ages  is  44  years.    Find  the  age  of  each. 

10.  The  perimeter  of  a  certain  square  is  160  feet.  Find 
the  length  of  each  side. 

11.  The  perimeter  of  a  certain  rectangle  is  216  feet.  It  is 
three  times  as  long  as  it  is  wide.    Find  its  dimensions. 

12.  The  perimeter  of  the  rectangle  formed  by  placing  two 
equal  squares  side  by  side  is  258  inches.  Find  the  side  and 
the  perimeter  of  each  square. 


INTKODUCTION  9 

5.  Literal  notation.  In  algebra  numbers  are  represented 
by  one  or  more  arable  numerals,  or  by  letters,  or  by  both 
combined. 

Thus  3,  25,  a,  2  b,  4  xi/,  and  2  a:  +  3  are  algebraic  symbols  for 
numbers. 

Precisely  what  numbers  4  xy  and  2x-{-  S  represent  is 
not  known  until  the  numbers  for  which  x  and  y  stand 
are  known.  In  one  problem  these  letters  may  have  values 
quite  different  from  those  they  have  in  another.  To  de- 
vise methods  of  determining  these  values  in  the  various 
problems  which  arise  is  the  principal  aim  of  algebra. 

6.  Factors.  A  factor  of  a  product  is  any  one  of  the 
numbers  which  multiplied  together  form  the  product. 

Thus  3  ab  means  3  times  a  times  b.  Here  3,  a,  and  b  are  each  fac- 
tors of  3  ab.  Similarly,  the  expression  4  (a  +  6)  means  4  times  the 
sum  of  a  and  b.   Here  4  and  a  ■{■  b  are  factors  of  4  (a  +  6). 

ORAL  EXERCISES 

1.  Name  the  factors  in  3  •  4  •  6,  2  xy,  3  abx^  4  abc. 

In  Exercises  2-5,  replace  a  by  3  and  ^  by  4  and  find  the 
value  of  the  resulting  expression. 

2.  ab.  3.  Sab.  4.  2 ab.  5.  5ab, 

7.  Exponents.  An  exponent  is  an  integer  written  at  the 
right  of  and  above  another  number  to  show  how  many 
times  the  latter  is  to  be  taken  as  a  factor. 

(Later  this  definition  will  be  modified  so  as  to  include 
fractions  and  other  numbers  as  exponents.) 

Thus  32  =  3  •  3  ;  5^  =  5  •  5  •  5.  Also  a^  =  a-  a*  a-  a,  and  4  xy^  = 
4: '  X '  y '  y '  y.  In  a*,  6  is  the  exponent  of  a.  If  a  is  4  and  b  is  3, 
a*  =  4^  =  4  •  4  •  4.   The  exponent  1  is  usually  not  written. 


10  riEST  COUESE  IK  ALGEBEA 

ORAL  EXERCISES 

1.  What  are  the  exponents  in  2aG^?  3  a^c  ?  5  aV  ? 

2.  What  is  meant  by  x^?  a^?  b'?  b'  ? 

3.  42  =  ?  4.  52  =  ?  5.2^=?  6.  3  .  52  =  ? 

In  Exercises  7-14,  replace  a  by  3  and  b  hj  2  and  find  the 
value  of  the  result. 

7.  a\  9.  a^  11.   a%  13.  2a\ 

8.  0^24.^2^  10.  ^3_,_^8^  12.  a^b^  14.  5  a^^^ 

8.  Coefficients.  If  a  number  is  the  product  of  two 
factors,  either  of  these  factors  is  called  the  coefficient  of 
the  other  in  that  product. 

Thus  in  4  x'^y,  4  is  the  coefficient  of  x^i/y  y  is  the  coefficient  of  4  x^, 
and  4  ?/  is  the  coefficient  of  x^.  The  numerical  coefficient  1  is  usually 
omitted.  If  a  numerical  coefficient  other  than  1  occurs,  it  is  usually 
written  first.    For  instance,  we  write  5  x,  not  x  5. 

The  following  examples  illustrate  the  difference  in  mean- 
ing between  a  coefficient  and  an  exponent  respectively: 

vfi  =  X  '  X  '  X. 

If  x=?)  in  each  case,  '^  x  stands  for  the  number  15, 
while  a?  stands  for  125.  If  x  =  10  in  each  case,  3  a;  =30, 
while  0.^  =  1000. 

ORAL  EXERCISES 

1.  What  are  the  numerical  coefficients  in  4cc?  5a^?  ^ax? 
Aac?  Sabc? 

2.  What  is  meant  hj  Sa?  4.x?  5c? 

3.  In  4  a^xy,  name  the  coefficient  of  a'^xi/j  xy,  y,  a^x,  and  a^y, 

9.  Use  of  parentheses  and  radical  signs.  If  two  or  more 
numbers  connected  by  signs  of  operation  are  inclosed  in 
parentheses,  the  entire  expression  is  treated  as  a  symbol 
for  a  single  number. 


INTRODUCTION  11 

Thus  3(6+4)  means  3  ■  10,  or  30;  (17  -  2) -f- (8  -  3)  means 
15-4-  5,  or  3;  (5  +  ly  means  12^  or  144:  and  Q(x  -{•  y)  means  six 
times  the  sum  of  x  and  y. 

As  in  arithmetic,  the  symbol  for  square  root  is  V  ,  and 
the  symbol  for  cube  root  is  v   . 

The  name  radical  sign  is  applied  to  all  symbols  like  the 
following:  V  ,  V  ,  V  ,  etc.  The  small  figure  in  a  radical 
sign,  like  the  3  in  v   ,  is  often  called  the  index. 

ORAL  EXERCISES 


'ind  the  value  of : 

1.  2(3  +  4). 

6.  V9  +  7. 

2.  4(7-2). 

7.   V3^  +  4^. 

3.  (4 +  3)  (5 -2). 

8.  -\/4(7-5). 

4.  (7 -2)  (8 +3). 

9.   -?/(5  +  3)(6  +  2), 

5.   VI. 

10.   V6^  +  8^ 

Note.  There  has  been  a  considerable  variety  in  the  symbols  for 
the  roots  of  numbers.  The  symbol  V  was  introduced  in  1544  by  the 
German  Stifel,  and  is  a  corruption  of  the  initial  letter  of  the  Latin 
word  radix,  which  means  **root."  Before  his  time  square  root  was 
denoted  by  the  symbol  B,  used  nowadays  by  physicians  on  prescrip- 
tions as  an  abbreviation  for  the  word  recipe.  Thus  ^5  would  have 
been  denoted  by  B;^5.  Some  early  writers  used  a  dot  to  indicate 
square  root,  and  expressed  V2  by  •  2.  The  Arabs  denoted  the  root 
of  a  number  by  an  arable  letter  placed  directly  over  the  number. 

ORAL  EXERCISES 

1.  What  are  the  numerical  coefficients  i\\2,x?  Sa^?  Axy? 
2ab?  S^a? 

2.  What  are  the  exponents  in  3a%?  4.a%^?  5aV?  5icV^? 

3.  What  is  the  difference  in  meaning  between  the  4  in  4  a; 
and  that  in  cc^  ? 

4.  What  is  meant  by  2 a; ?  5a?  8a? 


12  FIRST  COUESE  IN  ALGEBRA 

5.  What  is  meant  by  2  a^  ?  3  a^^  9 

6.  What  is  meant  by  3 (8 +  6)?  2(9-4)?  (7  +  3)(8~2)? 
(7  +  3)2?  V3  +  6?  V9+I6?  VlOO  -  64  ?  -y/SB  -  8 ? 
■^100  -  36  ? 

7.  What  is  the  numerical  value  of  each  expression  in 
Exercise  6  ? 

8.  Read  Exercises  1-16  on  pages  15-16. 

9.  3.52=?  13.  (5-l)(8  +  3)=? 

10.  (8  +  2)^=?  14.  3(7-2)(5-3)=? 

11.  7(6-1)=?  15.   V52+122  =  ? 

12.  (4 +  3)  (5 +  4)=?  16.   V(10  +  8)(10-8)  =  ? 

In  Exercises  17-33,  replace  a  by  3  and  J  by  4  and  find 
the  value  of  the  resulting  expressions. 

17.  Sa^-2b.  21.  2al  25.  Sa%\  29.  V3F+T6. 

18.  Sb^-\-Sa.  22.  2a%.  26.  4a5  +  ^l  30.  V4a2  +  7^>. 

19.  Vb.  23.  4a^>l  27.   V3a^.  31.  W^. 

20.  R  24.  2a''b\  28.   Va^  ^-  b\  32.  "V^a^^ +15. 

33.    -\/a2  +  52-17. 

EXERCISES 

Write,  using  algebraic  symbols  : 

1.  The  sum  of  three  times  a  and  four  times  b. 

2.  Three  times  a  subtracted  from  four  times  b.  , 

3.  The  square  of  a  subtracted  from  the  square  of  b. 

4.  The  cube  of  b  subtracted  from  the  square  of  a, 

5.  Two  times  a  squared  subtracted  from  three  times  a 
squared. 

6.  The  quotient  of  a  and  b. 

7.  The  product  of  four  times  a  squared  and  b. 


INTEODUCTION  13 

8.  The  sum  of  a  and  b  divided  by  their  product. 

9.  The  product  of  a  and  2h  —  c. 

10.  The  product  of  a  and  the  sum  of  b  and  c, 

11.  The  result  of  subtracting  a  —  b  from  Ix, 

12.  The  sum  of  the  square  root  of  5  a  and  the  cube  root 
of  lb. 

13.  The  product  oi  x  —  y  and  the  square  root  of  Ix. 

14.  The  square  of  the  sum  of  a  and  b. 

15.  The  square  of  b  subtracted  from  a. 

16.  The  quotient  of  three  times  a  multiplied  by  the  square 
of  b,  and  four  times  c  multiplied  by  the  cube  of  a. 

17.  The  sum  of  the  quotients  of  a  and  3x,  and  4y  and  c. 

10.  Order  of  fundamental  arithmetical  operations.  If  we 
read  the  expression  6  +  4.9  —  12-7-3  from  left  to  right, 
and  perform  each  indicated  operation  as  we  come  to  its 
symbol,  we  obtain  successively  10,  90,  78,  and  a  final 
result  of  26.  If  we  perform  the  multiplication  and 
division  first,  the  expression  becomes  6  +  36  —  4,  which 
equals  38.  These  results  show  that  the  value  of  the 
expression  is  determined  largely  by  the  order  in  which 
the  operations  are  performed.  It  is  customary  to  ob- 
serve the 

Rule,  In  a  series  of  operations  involving  addition^  subtrac- 
tion, multiplication,  and  division  of  arithmetical  numbers,  the 
multiplications  and  divisions  shall  be  performed  first^  in  the 
order  in  which  they  occur.  The  additions  and  subtractions 
in  the  resulting  expression  shall  then  be  performed  in  the 
order  in  which  they  occur  or  in  any  other  order. 

If  parentheses  occur,  each  expression  within  parentheses 
should  first  be  simplified  in  accordance  with  the  preceding 
rule  and  the  rule  then  applied  to  the  whole- 


14  FIRST  COUESE  IN  ALGEBEA 

EXAMPLES 
Simplify : 

1.  18  -f-  2  +  5  -  4  .  2. 

Solution.    18-^-2  +  5-4. 2  =  9  +  5-8  =  6. 

2.  24  --  8  .  4  -  6  +-  5  .  2  -  7. 

Solution.    24 -^8. 4-6  + 5- 2-7  =  3-4-6+ 10 -7  = 

12  -  6  +  10  -  7  = 

22  -  13  =  9. 

3.  4.  2- 3+- 2(8.  4 -12 -3 +  2- 6) -IS- 2. 

Solution.   4. 2-3  +  2  (8. 4- 12 -4-3  +  2- 6) -18 -4-2  = 

8-3  +  2  (32-    4+    2- 6) -9  = 

5  +  2  (  24  )  -  9  = 

5  +  48  -  9  =  44. 

EXERCISES 

Simplify  the  following : 

1.  20  -  5  -f  6  -  10.  6.  18  --  (2  .  3). 

2.  16  -  (8  -  2).  7.  (6  -  3)  .  (17  -  2  .  5)  . 

3.  14-(16-8)-f(12-4).    8.  23  -  2  •  6  -  4 -- 2 +- 16. 

4.  6 --3 -2.  9.  18-^(9-3). 

5.  8  .  6  -  3  -  10.  10.  (10-3)  .  (16-3  •  2+-8-5-4). 

11.  14  -  3  .  (16  -  2  .  5)  -  6  +-  8  .  2. 

12.  (18  -  2)  -  (4  +  2  .  8  -  18  -  9)  -  6. 

13.  (16  -  6)  .  (18  -  8)  -V- 100  .  5  -  5. 

14.  (5  -f  3)  .  (5  -  3)  -f-  4  -  3. 

15.  3^-2.  3-1 +-11  18.  8(12 +  4+- 3-1). 

16.  8^  -  2  .  8  .  3  +  3".  19.  3  +  2  .  4  +-  (3  +-  2)4.  c/ 
17.*  4.  3-2(6-2.3)4-8.        20.  (8-2)3  +  8-2-3. 

21.  3.4-6-O+-2.5+-2-72-2.8. 

22.  (12  -f  24  X  18  ^  3  +  6)  .  (24  --  4  +  3  -  2). 


INTEODUCTION  15 

11.  Evaluation  of  algebraic  expressions.  It  is  frequently 
necessary  to  find  the  numerical  value  of  an  expression  for 
certain  values  of  the  letters  involved.  This  process  will 
be  found  useful  in  detecting  errors  made  in  the  solution 
of  equations.    (See  page  42.) 

In  practical  affairs  a  working  rule,  a  geometrical  relation,  or  a 
scientific  fact  is  often  stated  briefly  and  conveniently  by  means  of 
an  algebraic  expression.  Such  expressions  are  frequently  called  for- 
mulas.  The  student  will  recall  that  arithmetic  furnished  many  illus 

trations  of  their  use.    Thus,  ^  =  —  is  the  formula  for  the  area  of  a 

triangle ;  I  =  P  -  r  -  t  is  the  formula  for  simple  interest ;  A  =  irr^  is 
the  formula  for  the  area  of  a  circle,  etc. 

In  finding  the  numerical  value  of  any  literal  expression 
the  student  should  observe  the  following 

Rule,  First,  put  in  place  of  each  letter  its  numerical  value; 
second,  simplify  the  result  thus  obtained. 

In  any  but  the  simplest  expressions  the  student  should 
always  observe  the  two  steps  of  the  above  rule  separately 
in  the  order  in  which  they  are  stated.  To  mix  the  two  in 
an  attempt  to  perform  mentally  both  processes  at  once,  is 
sure  to  result  in  many  errors  and  consequent  loss  of  time. 

EXERCISES 

In  Exercises  1-16,  let  «.  =  3,  5  =  1,  c  =  5,  tZ  =  7,  and  /=  2. 
Substitute  for  each  letter  its  numerical  value,  and  then  simplify 
the  results  according  to  the  rule  of  page  13  : 

1.  4.0" -Ih.  4.  6'2-3a5.  „    121)      c 

8.  —r-i' 

Solution.  5.  ahcd-hf,  J         ^ 

4a2-7&  =  4. 32-7.1  d  +  c  .    14.1^1 

-36-7  =  29.  ^-  ~7^'  c^d^f 

2.  4a  +  3d  7^^+^.  10.  -^V—. 

3.  ah  -f-  cd.  '   3/      2  a  '  ac       af 


16  FIEST  COUESE  IN  ALGEBRA 

^^    a^j^lP'j^o^j^d\  14.   af, 

12.  5/^+ 4/^- 4/- 5.  ^+/c 

13.  3/^ -9/^ +  11/^ -11/^ +  13/- 20.  ^^*  3c  +  cj* 

Find  the  numerical  value  of  the  following  expressions  when 
a  =  4,  Z>  =  0,  c  =  5,  c?  =  7,  and/=  8: 

^^    4a  +  3^>  +  2^  21.  cV2^.               27.    (a  +  c).a  +  c. 

C+/+2  22^  Vc^  -  a\             28.   a  +  c^  +  c. 

18-  a  +  \  +  d'  ^^-  (^+/).(c4-^).    29.   (/-a)l 

ah      hd      bf  24.  aK^  +  ^).           30.   (^  -  a)l 

^^'    c  "^  a  '^  cd  25.  (6^4-c).(a  +  c).   31.  a2-2ac  +  c2. 

20.   Va  +  V27.  26.  a  +  c(a  +  c).       32.  a'-\-2ah-^h\ 

33.  c^-3c2(i  +  3ca'-(t^  35.    Va^  +  ac. 

34.  d^JYaf.  36.    "v^acZ  +  36. 

37.  If  ^  =  2  and  y  =  3,  does  13a;  -^  5y  =11  ? 

38.  If  a;  =  8,  does  7a;-9  =  3x  +  25? 

39.  Does  0^2-503  +  6  =  0,  if  x  =  2?    ifx  =  3?    ifir  =  4? 

40.  Does  cc^-Ta; +12  =  0,  if  33  =  2?    if  a?  =  3  ?   if  ic  =  4  ? 

41.  Does  2a;2_5a;-3  =  0,  if  x  =  4?    if  a;  =  3  ?   if  a;  =  |  ? 

42.  The  area  of  a  triangle  is  given  by  the  expression  ^  =  — 

in  which  A  is  the  area,  a  is  the  altitude,  and  h  is  the  base.  Find 
the  area  of  a  triangle  in  which  the  altitude  is  11  inches  and  the 
base  14  inches. 

43.  The  area  of  a  circle  is  given  by  the  formula  A  =  irr^  in 
which  A  is  the  area,  tt  is  ^i^~  (approximately),  and  r  is  the  radius. 
Find  the  area  of  a  circle  whose  radius  is  (a)  7  inches ;  (b)  |-  inch. 

44.  The  volume  of  a  circular  cylinder  is  given  by  the  for- 
mula V=  TTT^h  in  which  V  is  the  volume,  r  is  the  radius  of  the 
base,  and  h  is  the  altitude  of  the  cylinder.    Find  the  volume 


IKTEODUCTION  17 

of  a  cylinder  in  which :  (a)  r  =  3  inches  and  h  =  5  inches ; 
(^)  r  =  4  inches  and  h=7  inches. 

45.  The  distance  a  body  falls  from  rest  is  given  by  the  for- 
mula  5  =  —  in  which  s  =  the  distance  in  feet,  a  =  32,  and  t  = 
the  time  in  seconds.   How  far  will  a  body  fall  in  5  seconds  ? 

46.  A  balloonist  drops  a  rock  while  crossing  a  river.  He 
sees  it  strike  the  water  16  seconds  later.  How  high  was  the 
balloon  at  the  time  ? 

47.  The  horse  power  of  a  certain  kind  of  gasoline  engine  is 

given  by  the  formula  H  =  -— -  in  which  H  is  the  horse  power, 

d  is  the  diameter  of  each  cylinder  in  inches,  and  n  is  the  number 
of  cylinders.  Find  the  horse  power  of  a  four-cylinder  engine 
in  which  the  diameter  of  each  cylinder  is  4  inches. 


CHAPTER  II 

POSITIVE  AND  NEGATIVE  NUMBERS 

12.  Addition  and  subtraction.  Let  us  suppose  that  equal 
distances  are  taken  on  a  line  and  the  successive  points  of 
division  are  marked  with  the  natural  numbers  as  follows : 

0123456789     10 
1 1 1 Lp— J I I I I I I '       '        ■         > 

(^) 

Such  a  scale  of  numbers  may  be  used  to  illustrate  both 
addition  and  subtraction  as  performed  in  arithmetic. 

Thus,  in  adding  5  to  3  we  may  begin  at  3  and  count  on 
5  spaces  to  the  right,  obtaining  the  sum  8.  We  shall  obtain 
the  same  result  if  we  begin  at  5  and  count  on  3  spaces  to  the 
right.    This  process  may  be  stated  in  general  terms  thus  : 

Rule,  To  add  the  number  a  to  the  number  b,  begin  at  b 
and  count  on  a  spaces  to  the  right. 

In  subtracting  4  from  7  we  may  begin  at  7  and  count 
off  4  spaces  to  the  left,  thus  obtaining  3.  This  process 
may  be  stated  in  general  terms  thus: 

Rule.  To  subtract  the  number  a  from  the  number  &,  begin 
at  b  and  count  off  a  spaces  to  the  left 

If  we  attempt  to  subtract  5  from  4  by  the  preceding 
rule,  we  arrive  at  the  first  point  of  division  to  the  left  of 
zero.  Arithmetic  has  no  number  to  represent  such  a  result; 
in  fact,  the  subtraction  of  5  from  4  is  there  regarded  as 
impossible.  Arithmetically  speaking,  such  a  subtraction 
cannot  be  performed.  We  can,  however,  subtract  4  of 
the  5  units  from  the  4  units,  leaving  1  unit  unsubtracted. 

18 


POSITIVE  AND  NEGATIVE  NUMBERS  19 

Now  in  algebra  it  is  both  convenient  and  necessary  to 
speak  of  subtracting  a  greater  number  from  a  less,  and  to 
call  the  portion  of  the  greater  number  which  is  unsub- 
tracted,  the  remainder.  The  fact  that  such  a  subtraction 
is  incomplete  is  indicated  by  writing  a  minus  sign  before 
the  result ;  thus,  4  —  5  =  —  1.  Hence  the  first  point  of 
division  to  the  left  of  zero  may  be  thought  of  as  corre- 
sponding to  —  1.  Similarly,  3  —  5  =  —  2  ;  and  to  —  2  may 
correspond  the  second  point  of  division  to  the  left  of  zero. 

In  like  manner  5  —  8  =  —  3,  which  corresponds  to  the 
third  point  to  the  left  of  zero.  In  the  same  way  the  fourth 
point  of  division  to  the  left  of  zero  would  correspond  to 
—  4,  the  fifth  point  to  —  5,  etc. 

Such  numbers  as  —  1,  —  2,  —  3,  etc.  are  called  negative 
numbers.  The  minus  sign  is  never  omitted  in  writing  a 
negative  number,  though  a  letter,  as  x^  may  denote  one. 

In  opposition  to  negative  numbers  the  ordinary  numbers 
of  arithmetic  are  called  positive  numbers.  If  a  number  has 
no  sign  before  it,  or  a  plus  sign,  it  is  a  positive  number. 

The  relative  order  of  positive  and  negative  numbers  is 
indicated  in  the  following  scale: 

-6  -5  -4  -3  -2  -1   0  +1  +2  +3  +4  +5  46  +7 
<»'■'■''' I I I I I I I I ^ 

(^) 
ORAL  EXERCISES 

Perform  the  following  additions  and  subtractions  by  count- 
ing along  the  preceding  scale  according  to  the  rules  on  page  18 : 

1.  Add  4  to  3.  3.  Add  6  to  ^  3.        5.  Add  2  to  -  5. 

2.  Add  4  to  -  2.        4.  Add  3  to  -  3,        6.  Add  5  to  -  7. 

7.  Subtract  2  from  5.  10.  Subtract  4  from  —  3. 

8.  Subtract  5  from  2.  11.  Subtract  2  from  —  4. 

9.  Subtract  6  from  3.  12.  Subtract  2  from  -  3. 


20  FIRST  COUESE  IN  ALGEBRA 

13.  Practical  use  of  positive  and  negative  numbers.  The 
scale  (^)  of  positive  and  negative  numbers  could  be  used 
to  measure  many  of  the  things  with  which  we  come  in 
daily  contact.  In  fact,  a  practical  equivalent  is  already  in 
use  in  many  instances.  Thus,  in  graduating  a  thermometer 
a  certain  position  of  the  mercury  is  taken  as  zero,  and 
the  degrees  are  marked  both  above  and  below  this  point. 
Hence  a  temperature  reading  of  18°  is  indefinite  unless 
accompanied  by  the  words  above  zero  or  behw  zero.  Usually 
+  18°  is  taken  to  indicate  the  former,  while  —18°  indicates 
the  latter. 

Similarly  any  point  on  the  earth's  equator  is  in  zero  lati- 
tude. Latitude  40°  N.  means  40°  north  of  the  equator. 
In  like  manner  30°  S.  means  30°  south  of  the  equator. 
Obviously  +  40°  and  —  30°  might  be  used  to  convey  the 
same  ideas. 

In  general  all  concrete  uses  of  positive  and  negative 
numbers  occur  in  connection  with  magnitudes  which  may 
be  regarded  as  opposite  in  sense;  as,  for  example,  money 
in  bank  and  an  overdrawn  account,  distances  measured  in 
opposite  directions  from  a  fixed  point,  and  time  measured 
in  the  future  and  in  the  past  from  a  certain  instant. 

EXERCISES 

1.  If  the  temperature  is  now  -f  14°,  what  will  represent  the 
temperature  after  a  fall  (a)  of  5°  ?  (b)  of  10°  ?  (c)  of  18°  ? 

2.  If  the  temperature  is  now  —16°,  what  will  it  be  after  a 
rise  {a)  of  7°  ?  {h)  of  12°  ?  (c)  of  25""  ? 

3.  In  the  preceding  exercise  change  the  word  rise  to  fall 
and  then  answer. 

4.  A  ship  sails  south  from  latitude  +13°  to  latitude  —7**. 
If  one  degree  is  69  miles,  how  far  did  it  sail  ? 


POSITIVE  AND  NEGATIVE  NUMBERS  21 

5.  A  ship  sails  south  from  latitude  +  20°  at  the  rate  of  5** 
daily.  In  what  latitude  is  it  at  the  end  of  each  of  6  days  ? 
After  how  many  days  will  it  reach  latitude  — 15°  ? 

6.  A  man's  property  is  worth  $4200  and  his  debts  amount 
to  $2300.  How  can  positive  and  negative  numbers  be  used  to 
represent  (a)  each  of  these  amounts  ?  (6)  the  man's  financial 
standing  ? 

Note.  So  far  as  is  known  the  first  explanation  of  positive  and 
negative  numbers  was  by  means  of  the  illustration  of  assets  and 
debts.  This  is  found  in  the  writings  of  the  Hindus  before  700  a.d., 
long  before  negative  numbers  were  accepted  as  having  any  definite 
meaning.  In  the  use  of  this  illustration  the  Hindus  were  nearly  a 
thousand  years  in  advance  of  the  times. 

7.  If  debts  and  property  be  reversed  in  Exercise  6,  what 
would  be  the  answer  to  (a)  and  (^)? 

8.  The  temperature  at  6.00  a.m.  was  —12°.  During  the 
morning  it  rose  at  the  rate  of  3°  an  hour.  What  was  the  tem- 
perature at  9.00  A.M.  ?  10.00  A.M.  ?  12.00  m.  ? 

14.  Addition  of  positive  and  negative  numbers.  As  we 
have  seen,  subtraction  by  the  use  of  scale  (B)  is  performed 
by  counting  spaces  to  the  left.  Now  a  negative  number 
represents  an  unperformed  subtraction ;  therefore  to  add  a 
negative  number  to  another  number  means  to  perform  this 
subtraction. 

For  example,  in  subtracting  8  from  5,-3  was  obtained 
by  beginning  at  5  and  counting  8  spaces  to  the  left,  arriv- 
ing at  3  to  the  left.  Hence  when  we  wish  to  add  —  3  to 
any  number,  we  count  off  3  spaces  to  the  left  from  that 
number. 

To  add  —  8  to  24  we  begin  at  24  and  count  off  8  spaces 
to  the  left,  obtaining  16  as  the  result ;  that  is, 

+  24  +  (-8)  =  +  16. 


22  FIEST  COUESE  IN  ALGEBRA 

Similarly,  to  add  —  6  to  —  4  we  begin  at  —  4  and  count 
6  spaces  to  the  left,  obtaining  —10  as  the  result;  that  is, 

-4 +  (-6)  =-10. 

Hence,  in  general,  to  add  a  negative  number  ti  to  a  given 
number,  begin  at  the  given  number  and  count  off  n  spaces 
to  the  left 

The  numerical  or  absolute  value  of  a  number  is  its  value 
without  regard  to  sign. 

Thus  the  absolute  values  of  —  3,  —  5,  and  +  7  are  3,  5,  and  7 
respectively.  It  should  be  noted  that  two  different  numbers,  as  +  6 
and  —  6,  may  have  the  same  absolute  value. 

ORAL  EXERCISES 

By  the  use  of  scale  (5),  page  19,  add  the  following  : 

1.  +  5,  +  3.     4.  -  5,  -  3.      7.  -  6,  -  2.     10.  -3,-5. 

2.  +5,-3.     5.  7,  -  3.  8.  6,  -  4.  11.  -  4,  +  4. 

3.  -  5,  +  3.     6.  -7,  -  3.       9.  4,  -7.  12.  -  6,  +  8. 

The  preceding  exercises  illustrate  the  correctness  of  the 
following  working  rules: 

/.  To  add  two  or  more  positive  numbers^  find  the  sum  of 
their  absolute  values  and  prefix  to  this  sum  the  plus  sign, 

II.  To  add  two  or  more  negative  numbers^  find  the  sum  of 
their  absolute  values  and  prefix  to  this  sum  the  minus  sign, 

III.  To  add  a  positive  and  a  negative  number^  find  the  dif- 
ference of  their  absolute  values  and  prefix  to  the  result  the  sign 
of  the  one  which  has  the  greater  absolute  value. 

These  rules  are  of  vital  importance,  as  they  are  of  almost 
constant  use  throughout  the  whole  of  algebra.  Rule  III  is 
the  one  in  the  use  of  which  errors  are  most  likely  to  occur. 


POSITIVE  AND  NEGATIVE  NUMBEES  23 

Hence  the  student  will  save  time  if  he  masters  Rule  III 
at  this  point. 

The  algebraic  sum  of  two  or  more  numbers  is  the  number 
obtained  by  adding  them  according  to  the  preceding  rules. 

The  algebraic  sum  of  two  numbers  is  often  different  from 
the  sum  of  their  absolute  values ;  for  example,  the  algebraic 
sum  of  +  9  and  —  5  is  +  4,  while  the  sum  of  their  absolute 
values  is  14. 

Hereafter  the  word  add  will  mean  find  the  algebraic  sum. 

ORAL  EXERCISES 

Perform  the  indicated  additions  : 

1.  +5 +(+7).  7.  -12 +(-9). 

2.  -5 +(-7).  8.  -6 +(+6). 

3.  +5 +(-7).  9.  -6 +(-6). 

4.  -5 +(+7).  10.  -4+(+3)  +  (+6). 

5.  +8 +(-5).  11.  3+(-7)  +  (+5)  +  (-4). 

6.  -8 +(+5).  12.  8  +  (-2)  +  (-4)-f(+6). 

Answer  the  questions  asked  in  the  following : 

13.  7  +  ?  =  9.  18.  -  8  +  ?  =  -  6. 

14.  7  +  ?  =  2.  19.  -10  +  ?  =-16. 

15.  8  -h  ?  =  12.  20.  -  10  +  ?  =  3. 

16.  8-f-?  =  5.  21.  12  +  ?  =  5. 

17.  _8  +  ?=-10.  22.  -12  +  ?  =  4. 

15.  Subtraction  of  positive  and  negative  numbers.   If  we 

wish  to  subtract  7  from  12,  we  may  do  so  by  answering 
the  question,  ''  What  number  added  to  7  gives  12  ?  "  By 
answering  a  similar  question  we  can  subtract  8  from  15, 
or  25  from  43,  or  any  number  a  from  another  number  6. 


24  FIEST  COUESE  IN  ALGEBEA 

Exercises  13-22,  p.  23,  are  therefore  exercises  in  subtrac- 
tion, for  each  asks  a  question  similar  to  the  one  in  the 
first  sentence  of  this  paragraph. 

This  point  of  view  brings  out  the  relation  that  the 
operation  of  subtraction  bears  to  that  of  addition. 

ORAL  EXERCISES 

Perform  the  following  subtractions  by  answering  in  each 
case  the  question,  ^^  What  number  added  to  the  first  number 
gives  the  second  number  ?  " 

Subtract : 

1.  5  from  9.  6.  -5  from  9. 

2.  9  from  14.  7.  5  from  —  10. 

3.  8  from  5.  8.-6  from  -  4. 

4.  13  from  7.  9.  12  from  -  18. 
5.-5  from  -  8.  10.  -  25  from  14. 

11.  In  Exercises  1-10,  change  the  sign  of  the  subtrahend 
(if  +,  to  —  ;  if  — ,  to  +)  and  then  add  the  subtrahend  to  the 
minuend.    Are  the  answers  the  same  as  were  obtained  before  ? 

The  results  obtained  in  Exercise  11  illustrate  the  fol- 
lowing important  principles: 

/.  Subtracting  a  positive  number  is  the  same  in  effect  as 
adding  a  negative  number  of  the  same  absolute  value. 

To  illustrate :  a  decrease  of  $100  in  a  man's  assets  is 
equivalent  to  an  increase  of  |100  in  his  liabilities,  pro- 
vided we  consider  his  real  financial  standing  in  each  case. 

//.  Subtracting  a  negative  number  is  the  same  in  effect  as 
adding  a  positive  number  of  the  same  absolute  value. 

To  illustrate:  a  decrease  of  $75  in  a  man's  liabilities  is 
equivalent  to  an  increase  of  $75  in  his  assets,  as  far  as 
his  net  financial  standing  is  concerned. 


POSITIVE  AND  NEGATIVE  NUMBERS  25 

Hence,  for  the  subtraction  of  positive  and  negative  num- 
bers, we  have  the 

Rule.  Ohange  the  sign  of  the  subtrahend  (tf+^  to  —;  if —^ 
to  +).  Then  find  the  algebraic  sum  of  the  subtrahend  (with 
its  sign  changed^  and  the  minuend. 

This  rule  really  turns  algebraic  subtraction  into  alge- 
braic addition  and  marks  one  of  the  most  important  dis- 
tinctions between  the  operations  of  arithmetic  and  those  of 
algebra.  For  this  reason  little  genuine  progress  in  algebra 
can  be  made  till  the  student  becomes  proficient  in  the  use 
of  the  method  stated  in  the  rule. 

ORAL  EXERCISES 

Subtract  the  second  number  from  the  first  in  Exercises  1-16 : 

1.  8,  +  5.       5.  14,  +9.  9.  +  6,  +  6.    13.  -15,  -19. 

2.  +8,-5.    6.  14,  -  9.        10.  -  6,  -  6.    14.  0,  +  1. 

3.  +5,  -8.    7.   -  14,  +  9.    11.  6,  -  6.         15.  -1,-1. 

4.  4-5,  +8.    8.  -  14,  -  9.    12.  -  ^^  Ji^  ^.    16.  1,  -  2. 

17.  12- (+3) -(+2)  =  ?  19.   _12-(-3)-(-2)=? 

18.  -10 -(+3) -(+2):=?        20.  17 -(-5)- (+7)=? 

Supply  the  missing  numbers  in : 

21.  +7  +  ?  =  10.  29.  9-?  =  5. 

22.  -6  +  ?  = -10.  30.  -7-?  =  - 5. 

23.  -5  +  ?  =  0.  31.  +5-?  = -9. 

24.  -f  6  +  ?  =  0.  32.  _  7  -  ?  =  4. 

25.  +4  +  ?  =  4.  33.  _5_?=:a 

26.  -  8  +  ?  =  -  3.  34.  2  -  ?  =  0. 

27.  -  9  +  ?  =  -  5.  35.  4  -  ?  =  16. 

28.  -  7  +  ?  =  7.  36.  6  +  ?  =  -  3. 


26  FIRST  COUESE  IN  ALGEBEA 

Simplify : 

37.  12  +  (7)  -  (5).  40.  18  +  (-  6)  -  (+  7). 

38.  12  -  (-  3)  +  6.  41.  - 16  -f  (- 10)  -  (+ 12). 

39.  12  -  (-  4)  +  (-  5).  42.  - 13  -  (8)  +  (- 14). 

16.  Multiplication  of  positive  and  negative  numbers.    In 

arithmetic,  multiplication  was  defined  as  the  process  of 
taking  one  number,  the  multiplicand,  as  many  times  as 
there  are  units  in  another  number,  the  multiplier.  The 
original  signification  of  times  made  this  definition  mean- 
ingless when  the  multiplier  was  a  fraction ;  for  in  8  x  f , 
8  could  not  be  added  |^  of  a  time.  The  definition  was 
then  extended  and  the  product  of  8  multiplied  by  |^  was 
defined  to  mean  8  multiplied  by  3  and  the  result  divided 

by  5 ;  that  is,  8  x  -  means  — 

^  5  5 

Since  algebra  deals  with  both  positive  and  negative  num- 
bers, we  must  now  extend  the  arithmetical  definition  of 
multiplication  and  define  what  sign  the  product  shall  have 
in  each  of  the  four  cases  which  may  possibly  arise: 
(+4).  (+3)=?  (+4).  (-3)=? 

(_4).(+3)  =  ?  (_4).(-3)=? 

By  (+  4)  .  (4-  3)  we  mean  that  +  4  is  to  be  added  three 
times:  (+ 4)  +  (_^  4)  +  (+ 4)^  +  12; 

that  is,  (+  4)  .  (+  3)  =  -f  12. 

Similarly  (—  4)  .  (+  3)  means 

(_4)  +  (-4)  +  (-4)=:-12; 
that  is,  (-4)  .(+3)  =-12. 

In  (+  4)  .  (—  3)  we  mean  that  4  is  to  be  subtracted 
three  times.    This  is  the  same  as  subtracting  12  once. 

Therefore  (+  4)  .  (-  3)  =  - 1 2. 


POSITIVE  AND  NEGATIVE  NUMBERS  27 

Lastly,  by  (—4)  .  (—3)  we  mean  that  —  4  is  to  be  sub- 
tracted three  times.  This  is  the  same  as  subtracting  —12 
once,  and  subtracting  —  12  once  is  the  same   as  adding 

+  12.    Therefore 

(_4).(-3)  =  +  12. 

Summing  up, 

(+4)  .  (-j-3)  =  +  12.  (+4)  .  (-3)  =  -12. 

(-4)  .  (+3)  =  -12.  (-4)  .  (_3)  =  +  12. 

Or,  in  general  terms, 

■j-  a  x-\-h  =  -\-  ab. 

—  a  X  +  b  =—  ab. 
■i-  a  X  —  b  =  —  ab. 

—  a  x  —  b=-\-  ab. 

Therefore  we  have  the 

Rule.  The  product  of  two  numbers  having  like  signs  is  a 
positive  number,  and  the  product  of  two  numbers  having  unlike 
signs  is  a  negative  number. 

ORAL  EXERCISES 
Eind  the  products  of  the  following : 

1.  +  3,  +  5.  7.   -  12,  +  9.  13.  4-  4,  -  7,  -f  6. 

2.  +  4,  +  12.  8.   +  6,  -  4.  14.  +4,-5,-6. 

3.  -  5,  +  6.  9.   -  7,  -  6.  15.  -4,-5,-3. 

4.  +  6,  -  6.  10.  +  5,  -  10.  16.  12,  +0,-5. 

5.  -  7,  +  8.  11.  +  0,  +  4.  17.  9,  -  10,  -  0. 

6.  -  7,  -  4.  12.   -  7,  0.  18.  -  4,  +  7,  -  6. 

19.  -3,-2,-5.  20.  2,  -  3,  +  9. 

Note.  The  famous  German  mathematician  Leopold  Kronecker 
(1823-1891)  once  observed  that  "the  good  Lord  made  the  positive 
integers,  but  man  is  responsible  for  all  the  rest  of  the  numbers." 


28  FIRST  COURSE  IN  ALGEBRA 

This  expresses  the  truth  about  numbers  as  accurately  as  one  can  in 
a  single  sentence.  We  count  objects  from  our  earliest  years,  and  so 
use  the  positive  integers  naturally.  It  is  only  when  we  come  to  study 
mathematics  that  the  necessity  for  any  other  kind  of  numbers  is 
forced  upon  us.  Here  we  see  that  negative  numbers  are  a  great  con- 
venience if  we  wish  to  represent  the  relations  between  objects  where 
oppositeness  in  any  of  its  many  forms  is  involved.  But  the  artificial 
character  of  negative  numbers  delayed  their  intelligent  use  for  many 
hundred  years.  To  be  sure,  the  Hindus  said  that  "the  square  of 
negative  is  positive,"  but  the  statement  probably  did  not  mean 
anything  to  those  who  read  it.  It  was  not  until  after  the  time  of 
Descartes  (see  p.  210)  that  the  rules  for  operating  on  negative 
numbers  were  understood,  even  by  great  mathematicians. 

17.  Division  of  positive  and  negative  numbers.   When  18 

is  divided  by  9  the  result  is  2.    Here  18  is  the  dividend, 

9  the  divisor,  and  2  the  quotient.    The  three  are  connected 

by  the  following  relation,  which  holds  both  for  arithmetic 

and  alsjebra :  .  _  .  , .   _      _ 

quotient  x  divisor  =  dividend. 

We  can  see  that  2  is  the  correct  value  of  18-^-9,  because 
2  X  9  =18.  This  simple  test  will  be  applied  to  determine 
whether  the  quotient  is  a  positive  or  a  negative  number. 
All  the  cases  which  may  arise  are  represented  by  the  four 
following  questions: 

(a)     +18-^+9  =  ?  ((?)     +18-v--9  =  ? 

(h^    _18--.+  9  =  ?  (d)    -18-^-- 9  =  ? 

These  questions  are  answered  as  follows : 


(a) 

+  18- 

-+9=  +  2  because  +2 

+  9  =  +  18. 

(J) 

-18- 

-+9=-2  because  -2 

+  9=-18. 

(<^) 

+  18- 

—  9  =  —  2  because  —  2 

-9  =  +  18. 

id) 

-18- 

--9  =  +  2  because  +2 

-9  =  -18. 

In  (a)  and  ((^)  the  dividend  and  divisor  have  like  signs 
and  the  sign  of  the  quotient  is  plus.    In  (5)  and  (tf)  the 


POSITIVE  AND  NEGATIVE  NUMBERS  29 

dividend  and  divisor  have  unlike  signs  and  the  sign  of  the 
quotient  is  minus. 

Therefore  we  have  the 

Rule,  The  quotient  of  two  numbers  having  like  signs  is  a 
positive  number^  and  the  quotient  of  two  numbers  having  unlike 
signs  is  a  negative  number. 

The  result  of  multiplication  by  zero  is  given  a  definite  meaning 
in  arithmetic  and  algebra,  namely  zero ;  but  in  both  subjects  division 
by  zero  is  always  excluded.  If  zero  were  used  as  a  divisor,  numerous 
contradictions  would  arise  of  which  the  following  is  an  illustration : 


Obviously, 

0-4  =  0, 

and 

0-6=0. 

Therefore 

0-4  =  0-6. 

Dividing  each  by  zero, 

4  =  6, 

which  is  false. 

Note.  The  Hindus  were  the  first  to  express  the  laws  that  govern 
the  operations  with  the  number  0.  In  fact,  they  were  the  first  to 
have  such  a  symbol.  In  the  twelfth  century  a  Hindu  writer  states 
that  a  +  0  =  a,  that  Vo  =  0,  and  that  0^  =  0.  Of  course  he  did  not 
express  himself  in  terms  of  these  symbols,  but  in  the  notation  of 
his  time  and  country. 

ORAL  EXERCISES 

Divide  the  first  number  by  the  second  in  Exercises  1-9 : 

1.  +10,  +  2.  4.  +14,  -  2.  7.  0,  +  5. 

2.  -10,  -  5.  5.  -18,  -  3.  8.  0,  -  5. 

3.  _15^  4-  3.  6.  -7,  +7.  9.  -  3,  -  3. 


(-2)  =  ?         16.  48 --2 -(-4)=? 
(-3)  =  ?         17^  1^20^4 


10.  +36  ^(+3) 

11.  4_45^(_5) 

12.  -64^(+4)-..(-2)  =  ? 

13.  +96--(-6)-(+8)=:?  18.  -^=-2. 

14.  72^(+6)^(-4)=?  ^'3(5  _ 

15.  60-r-(-5)H-(-12)=?  ^^-  "T~~ 


30  FIEST  COURSE  IN  ALGEBEA 

18.  Omission  of  the  plus  sign  before  a  number.    If  the 

first  of  several  numbers  connected  by  either  plus  or  minus 
signs  is  a  positive  number,  its  sign  is  omitted;  thus 
+  4—3  +  6  is  written  4  —  3  +  6.  If  the  sign  of  the  4  had 
been  negative,  the  minus  sign  could  not  properly  have  been 
omitted. 

19.  Omission  of  the  sign  of  multiplication.  If  each  of 
two  or  more  numbers  be  inclosed  in  a  parenthesis  and  the 
parentheses  be  written  one  directly  after  the  other  with 
no  sign  of  operation  between  them,  the  sign  of  multipli- 
cation is  always  understood ;  thus  (6)  (3)  means  6  •  3. 
Similarly  (6)  (5)  (2)  =  6  .  5  .  2. 

MISCELLANEOUS  ORAL  EXERCISES 

Simplify  the  following : 

1.  (7) +  (5).               13.  12-18.  25.  5-0. 

2.  (7) -(5).                14.  -18-12.  26.  0  •  (- 9). 

3.  (7)  +  (-  5).           15.  15  -14.  27.  4  •  8. 

4.  (7)_(_5).           16.  +7-0.  28.  -3.6. 

5.  (_7)_f-(5).           17.    0-3.  29.  12 --(-2). 

6.  -7+5.                 18.  (-3) (6).  30.  -12 --2. 

7.  (_9)_(4).           19.  (-5)6.  31.  -39-^(-3). 

8.  -9-4.               20.  (7)  (-  5).  32.  45  ^  (- 15). 

9.  -11  + (-13).      21.  8 (-3).  33.  0-^(-6). 

10.  _6-(-10).        22.  (-5) (-12).  34.  0-^3. 

11.  -6+10.              23.  -3(-8).  35.  -27-^9. 

12.  8  +(-10).            24.  -  5  .  4.  36.  3  -  5  +  6. 
37.  _4_j-6-2  +  l.           38.-4  +  6  +  2-1. 

39.  2-3  +  4-5-6. 


POSITIVE  AND  NEGATIVE  NUMBERS  31 

Add: 

42.  -8  43.       4 

6  -9 

2  -3 

-5  6 


52.  (-2)1 

53.  (-4)^+ (4)1 

54.  -(-5)1 

55.  -(-5)1 

56.  -2  (3)2  (-2). 

57.  _3(_2)(-3) 

58.  -2 (4)2  (-5). 

59.  2(-4)2.5. 


40.       7 

-2 

3 

-5 

41.        6 

-2 

-3 

4 

Simplify : 

44.  3  •  6  H-  3. 

45.  _4(7)-f-(- 

46.  3  (-6)  ^2. 

47.  4.6(-8)- 

48.  18 --(-3)  • 

-2). 

.(-16). 
6-^4. 

49.  31 

50.  (-3)". 

51.  2». 

MISCELLANEOUS  EXERCISES 

Simplify  the  following : 

1.  4^-  (-  2)\  4.  (6  _  2)(7  -  3)^(4  -  9). 

2.  3'-(-2)».  5.  (6- 1)  (5+1). 

3.  (5-3)(3  +  2).         6.  (-l)^+(-l)»+(-2)^  +  (-2)» 

7.  9 +  3.  2 +  18 -=-(-3).         9.  3-6-s-9-2.6-5-4  +  (-3)'. 

8.  62_4h-(-2)-^6(-3).        10.  3^+2.3-(-4)-5-(-4)^ 

11.  (+2)^- 2(2)  (-3)  + (-3)1 

12.  3(4)''(-6)-(-6)^ 

13.  3'  +  3  (3)'^(-  2)  +  3  (3)  (-  2^  +  (-  2)». 

14.  4^-3 (4)X-  3)  +  3 (4)  (-  3)''-  (-  3)'. 

15.  3»-3(3)2(0)+3(3)(0)^-0'. 

16.  5^+3  (5)\-  4)  -f  3  (5)  (-  4)^  -  (-  4)». 


32  riKST  COURSE  IN  ALGEBRA 

If  ic  =  3  and  y  =  —  2,  find  the  value  of : 


17.  f,                 19.  y\ 

21.  2 1/2.                  23.  5ccV*. 

18.  y\                 20.  y\ 

22.  2?/l                  24.  4^y. 

25.  x^-if. 

28.  x2-2xi/  +  2/'. 

26.  x'-f. 

29.  (x^-y){x-y). 

27.  x^^'lxy^y'. 

30.  a;^-i-3ccV4-3V  +  ^'- 

31.  Does  4ic  - 

-2: 

=  2a:  +  8,  if  cc  =  5? 

32.  Does  3  X  - 

-5 

=  2aj  + 8,  if  x=-9? 

33.  Does  ic2_ic-12  =  0,ifa;  =  4?  ifcc  =  -8?  ifx  =  -4? 

34.  Does  3x2_|_i9^^;j^4^  if  ^^  2^  if  cc  =  2  ?  ifa^^-T? 

35.  At  7.00  A.M.  on  a  certain  day  the  thermometer  registered 
27  degrees  above  zero.  The  mercury  then  fell  at  the  rate  of 
3  degrees  per  hour.  What  was  the  temperature  at  noon?  at 
4.00  P.M.  ?  at  5.00  P.M.  ?  When  was  the  temperature  —  12  ? 

36.  A  Zeppelin  rises  1800  feet  from  its  position,  then  falls 
1200  feet,  rises  again  1750  feet,  falls  400  feet,  and  then  rises 
820  feet.    How  many  feet  higher  or  lower  is  it  than  at  first  ? 

37.  Euclid  lived  about  300  b.c.  Sir  Isaac  ISTewton  died  in 
1727  A.D.  If  dates  before  Christ  are  considered  negative  and 
those  after  Christ  are  considered  positive,  how  might  these 
dates  be  written  ? 

38.  What  is  the  meaning  of  the  date  -  450  ?  of  +  1917  ? 
What  is  the  difference  in  time  between  the  two  ? 

39.  A  boat  is  traveling  12  miles  per  hour.  A  man  on  its 
deck  is  walking  3  miles  per  hour.  Using  positive  and  nega- 
tive numbers,  represent  the  rate  at  which  he  approaches  his 
destination  when  he  walks  toward  the  bow  and  when  he  walks 
toward  the  stern. 

40.  A  balloon  capable  of  supporting  500  pounds  is  held  down 
by  10  men  whose  average  weight  is  150  pounds.  Using  posi- 
tive and  negative  numbers,  represent  the  weight  of  the  balloon, 
of  the  men,  and  of  the  balloon 'and  the  men  together. 


CHAPTER  III 

ADDITION 

20.  Monomials.  A  number  symbol  which  is  not  the  indi- 
cated algebraic  sum  of  two  or  more  number  symbols  is 
called  a  term  or  monomial. 

Thus  5,  —  a,  b\  a^Xj  and  —  4  cy^  are  terms.  Frequently,  where 
no  confusion  would  arise,  expressions  like  (a  +  &),  3  (x  —  y),  5  V?, 
and  Va  —  x  are  called  terms,  for  in  such  cases  one's  thought  is 
centered  not  on  the  parts  of  which  the  expression  is  composed  but 
on  a  single  number  for  which  the  whole  stands. 

21.  Similar  terms.  Terms  that  are  alike  in  every  respect 
except  their  coefficients  are  called  similar. 

Thus  3,  —  7,  and  9  are  similar  terms,  as  well  as  V2  and  3  v2.  Also 
a,  4  a,  and  — 10  a  are  similar  terms,  as  are  a^x,  —  3  a^x,  and  7a^x. 

22.  Addition  of  similar  terms.  We  have  already  learned 
that  6a  +  3<x  =  9a,  6a+(—  3a)=3a,  and  5xy-\-6x^  =11  xy. 
In  like  manner  the  sum  of  8  ^,  —  3  y,  2  y,  and  —  i/  is  6  ?/. 
The  terms  —  y,  a;,  ay,  and  —  e^x  are  equivalent  to  —  1  y, 
-{-Ix,  -[-lay,  and  —Ic^x  respectively. 

Thus,  for  adding  similar  terms  we  have  the 
Rule,    Find  the  algebraic  sum  of  the  numerical  coefficients 
and  prefix  this  result  to  the  common  literal  part 

ORAL  EXERCISES 

Find  the  sum  of : 

1.  +8,  -4.         5.  +12 a, -7a.     9.  +5x, -7x. 

2.  -f-9,  -12.       6.  -9a, +3a.    10.  -lice, +14 a;. 

.    3.-6,-9.         7.  -2a,+3a.     11.  -12ac, -4ac, +9ac. 
4.  —6  a, —4  a.    8.  +  8x,  —  3  a;.    12.  +  aa;,  —3  ax, +5  ax. 


34  FIEST  COURSE  IN  ALGEBEA 

13.  +  4  ax^  —  ax,  —  7  ax. 

14.  +  6  ax,  —  aXj  —  3  ax,  +  ax, 

15.  +  4  ac,  —  3  ac,  + 12  ac,  —  8  ac. 

16.  -\-Sac,  — 12  ac,  — 10  ac,  -\-15  ac. 

Combine : 

17.  ax  —  2 ax  -{-  Sax  —  3 ax. 

18.  4  am  —  6  am  — 10  am  +  8  am. 

19.  5  (xm  —  3  am +12  am —7  am. 

20.  6  am  —  7  am  +  3  am  —  2  am. 

21.  —  3  ao?  —  4  ace  +  7  ax  — 12  ax. 

22.  —  ac  +  4  ac  +  ac  —  2  ac  + 12  ac. 

23.  4  ao:;  — 10  ace  — 12  ace  +  7  ax. 

24.  Sax  —  5  ax  -\-7  ax  —  9 ax  -\-  8  ax. 

23.  Order  in  adding  terms.  Obviously  2  +  3  =  3  +  2,  and 
2-3  +  5  =  2  +  5-3=-3  +  5  +  2,  etc.  This  illustrates 
the  law  that  in  addition  the  terms  may  be  arranged  and 
added  in  any  order.  Hence  6  d-\-7  c  =  7  c-{-6  d,  and  the 
sum  of  3  and  x  is  either  ^  +  3  or  3  +  a: ;  also  a  +  6  =  5  +  a, 
and  a  +  J  +  c  =  6  +  c  +  a  =  c  +  a  +  J. 

24.  Addition  of  dissimilar  terms.  Algebraic  expressions 
for  the  sum  of  two  terms  which  are  not  similar,  such  as  6  c? 
and  7  c,  are  obtained  by  writing  them  one  after  the  other 
with  a  plus  sign  between  them ;  thus,  6  cZ  +  7  c.  The  addi- 
tion of  6  c?  and  —  7  c  is  indica-ted  by  writing  6  c?  +  (—  7  c), 
or,  more  briefly,  6  d  —  7  c.  Similarly  the  sum  of  Sx,  —  2  ?/, 
and  —7z  may  be  written  3  a;  +  (—  2  ^)  +  (—  7  ;2),  or,  omitting 
the  parentheses  and  the  unnecessary  signs,  Sx  —  27/—7z. 

Thus,  for  adding  dissimilar  terms  we  have  the 

Rule.    Write   the   terms   one    after   another  in   any  order^ 

giving  to  each  its  proper  sign. 

If  similar  and  dissimilar  terms  are  to  be  added,  the  two 

preceding  rules  must  both  be  applied. 


ADDITION  35 

EXERCISES 
Find  the  sum  of : 

1.  a,  4:b,  —  c.  4.  5  x^i/,  —  5  xy^,  c^y,  —  2  cy^, 

2.  4.x,  -2b,  5y,  10.  5.  6x,  -3a,  2b,  -5x,  3y. 

3.  3  ab"",  2 bx,  -  cy,  4:  a%.         6.  5a,  -  Ab,  +  Sc,  7 b,  -  2  (y". 

7.  Sa%+2b%-7c%-4:b%5a\ 

8.  5  a%  -5  ab%  ~  3  a%  3  a^^^  4  a^^^,  2  a^^l 
Simplify : 

9.  11+ 5  _9_  3+16- 25. 

10.  16  ac  —  9  a??/  +  5  ac  —  2  0??/  —  6  ac  + 11  cc^'. 

11.  7x-\-6a—15a  —  Sx  +  3x. 

12.  122/-172/4-10^>  +  20y-25&. 

13.  4:  ab  —  S  xy  —12  ab  -{-15  ab  —  xy. 

14.  14a;2-13/  +  ir2_5^2_^^2^ 

15.  5^2-4y2_11^2  +  ^2_7^2^ 

16.  7c2^  -  5a26  +  60^6^  ^  a^^  +  9  a^^. 

17.  -  4  ^2^,2  _|_  6  ^2^  _  15  ^2^2  +  3  ccy  +  0  a% 

18.  -Z»^-19a+17^>«  +  ^>'-0^^+13«. 

19.  12  a^b  +  6c'd-  a%  + 16  a%  - 13  a«^  -  25  c'd, 

20.  llV^-16V^  +  2lV^-Va. 

21.  3  Vaj  —  y  —  Vx  —  2/  +  8  Va;  —  y  —7  Vic  —  y. 

22.  3(a  +  ^)-5(a  +  ^)+8(a  +  ^). 

23.  _7(a-2Z>)  +  (a-2^>)4-12(a-2^>). 

25.  Addition  of  polynomials.  A  polynomial  is  an  algebraic 
expression  consisting  of  two  or  more  terms. 

It  is  not  usual  to  call  an  expression  a  polynomial  if  any  of  its 
terms  contain  a  letter  under  a  radical  sign.  Thus  we  shall  not  caU 
expressions  like  Va;  —  3  +  4  polynomials. 

A  binomial  is  a  polynomial  of  two  terms. 
A  trinomial  is  a  polynomial  of  three  terms. 


86  FIRST  COURSE  IN  ALGEBRA 

EXAMPLE 

Add  the  following  polynomials  :  4  a  —  6  ^  —  a^c ;  3  Z>  +  4  a^c ; 
-Sa-Ja'c  +  lO',  5a-^Sb~6. 

Solution.  -i-  4  a  —  6  &  —     a^c 

+  3  &  +  4  a^c 
-3a  -  7 a^c  + 10 

+  5a  +  3& -    6 

Sum,        Q  a  —  4  a'^c  +    4 

For  the  addition  of  polynomials  we  have  the 

Rule.    Write  similar  terms  in  the  same  column. 
Find  the  algebraic  sum  of  the  terms  in  each  column  and 
write  the  results  in  succession  with  their  proper  signs. 

A  check  on  an  operation  is  another  operation  which  tests 
the  correctness  of  the  first. 

For  example,  in  arithmetic  the  result  of  division  is  checked  by 
multiplication ;  thus  the  check  for  132  -^  6  =  22,  is  22  •  6  =  132. 

Addition  in  arithmetic  is  usually  checked  by  adduig  the 
columns  in  the  opposite  direction.  In  the  two  cases 
the  mind  deals  at  every  step  with  different  numbers,  and 
if  the  final  result  is  the  same,  its  correctness  is  more 
probable  than  after  either  addition  alone.  But  for  arith- 
metical addition,  and  for  many  of  the  operations  of 
algebra,  no  really  satisfactory  check  exists. 

In  elementary  algebra  long  columns  in  addition  do  not 
occur.  Where  columns  contain  three  or  more  numbers 
they  may  be  added  term  by  term  upward  or  downward. 
If  the  numbers  in  a  column  are  not  all  of  the  same  sign, 
we  can  partially  check  results  by  adding  the  positive 
and  the  negative  numbers  in  such  columns  separately  and 
finding  the  algebraic  sum  of  the  two  results. 


ADBITIOK  87 

EXERCISES 

Add  the  following  polynomials  and  check  the  results : 

1.  X  -{-  S,  2x  —  6,  and  Sx  —  5. 

2.  X  —  y  —  S,  2x  +  y  -  S,  2ind  7 X  -  4:y  -\- 10. 

3.  x-^y  +  z,  X  —  3y-{-4:Z,  and  3i»-h4y  —  7«. 

4.  2x-\-5y  —  Zj3x  —  Sy-{-6z,  and  x  —  y  —  z. 

5.  3a; -1,4- 2x,  6a; -8,  and  3^ +  3. 

6.  x^-  9,  16  -  ^x^,  Sx''+  4,  and  3  -  2a;l 

7.  5x-^5y,  4:X  —  9y-\-6z,  and  Sx  —  Sy  —  3z. 

8.  Tec-  62/  +  3^,  5ic  —  4^,  and  2i:c  +  62/  —  5«. 

9.  a;2_i^_2,  3ic2-4a;  +  5,  andSx^+Sx-T. 

10.  a;^-  4a;  +  1,  3a;  -2x^-\-  8,  and  10a;  -  11  a;^-  18. 

11.  3  c  -  4  c^-  5,  c2+  c  +  1,  4  c  -  2  c2+  3,  and  5  -  2  c  -  5  <^, 

12.  4:X  —  5y  —  5z,  6y  —  2z,  and  7x  —  6y  —  9z. 

13.  x-{-2z-\-Sy,  y  —  Sz-{-x,  and  «  —  2  a;  —  4 1/. 

14.  5a;  —  6y  +  7^,  2y  —  Hz  -{-  Xy  and  9 «  —  5 1/. 

15.  8a  —  7^  —  6c,  5c  —  4a  —  3^>,  and  3 &  +  7 c. 

16.  9  ac  —  Z>c,  7  ab  —  3  ac,  and  —  12  a^  —  ac. 

17.  a2_4a  +  10,  5a  -  6^^+ 4,  and  3a  -  16  +  2a2. 

18.  9  -  a2+  3  a,  -  13  +  6  a"-  4  a,  and  3  a^-  a. 

19.  a  — a;  — 3c,    4  — 7 a; +  3 a,    7c  — 2a +  10,    and   a  — c 
-2a; -11. 

20.  a  +  a;  +  c  +  5,  2a  +  2c  +  10-|-2a;,  a  —  c  —  7  —  aj,  and 
7  —  3a;  —  2a  +  4c. 

21.  a;  —  1  —  3c,  a;  —  a  —  2c,  a  —  3c  —  5,  and  2x  —  7  -{-11a, 

22.  2a;  +  32/  +  5^,  a;-?/  +  3«,  and  3x  —  2y+10z, 

23.  4a;-20^,  8a;  +  9y,  and5^-3y. 

24.  a-3(a;-?/)+^,  5— 10a+4(a;  — 3/),and-2(a;-3/)-f  6. 

25.  a-{'d-{-2(b-c),  7(b-^c)-\'6d-12a,  and  lla-5(&-c). 


38      '  FIKST  COUESE  IN  ALGEBEA 

Combine  similar  terms  in  the  following  polynomials : 

26.  7a^-lSb^-\-  12  c^  -{.15b^  -  a"  -  Tc"  +  Sb^  +  5 c\ 

27.  x"  -\-2xy  -\-  y''  -  6xij  -  A.x'  -  y''  -{-Ux"  -  %xy  ^  y\ 

28.  5  ab  -  a""  -^  b^  -{-  4.ab  -  9b''  -\-  ^  a"  -  2  ab  -  2b\ 

29.  5^2- 6a;  +  ll-4x-8-3icM-Ta;-18  +  13cc. 

30.  12 c^  -  lObc  +  8^2  -f  ^c  -  6^»2  -  c^  +  c'  -  11  c\ 

31.  4 irV  —  i«2/  +  2/^—  ^ ^2/  +  4: J^2/^  —  2  x^y  +  4y2-|_  3^^  _j^ ^2^^ 

32.  la-^\b-\c  +  c  +  \b-la-b-\-lc  +  \a  +  l. 

The  sum  of  5  x  and  2  x  may  be  written  (5  +  2)  x.  *  This  is  not  usual  or 
necessary,  as  5  and  2  can  be  combined  and  the  result  written  7  x.  In 
adding  5  x  and  ox,  however,  the  5  and  the  a  cannot  be  combined  and  the 
result  expressed  by  a  single  character,  so  the  sum  is  written  (6  +  a)x. 
Similarly,  ax  —  3  x  =  (a  —  3)  x,  ax  ■\-  x  =  {a  +  l)x^  and  ax  —  6x  +  x  = 
(a  -  6  +  l)x.   Also  a(x  ^  y)  +  h {x  ■{■  y)  =  {a  +  h)  (x  ■\-  y). 

Write  so  that  x  will  have  a  polynomial  coefficient : 

33.  ax  +  Sx.        35.  Scx  —  x,  37.  2  aa;  —  3  a;  +  5cc, 

34.  2ax  -\-  X.        36.  aa?  +  5a?  +  caj.        38.  S  ax  —  A  ex  -]-  x. 
39.  3 aic  —  ^a?  —  X  +  a^o:;.         40.  bx  —  5 ex  —  x  —  4cbx. 

Write  so  that  the  binomial  will  have  a  polynomial  coefficient: 
41.  a(b-{-e)-{-S(b-{-c).  42.  4.(a  -  x)- 5b(a  -  x). 

43.  Sa(a-\-Sb)-l(a  +  3b). 
,    44.  76(x2  +  2/')-a(aj2  +  i/2)  +  (aj2  +  2/')- 


CHAPTER  IV 

SIMPLE  EQUATIONS 

26.  Definitions.  An  equation  is  a  statement  of  equality 
between  two  equal  numbers  or  number  symbols. 

Thus  2  =  5-- 3,  a-2b  =  ^a  +  h-2a-3b,  4a;  =  a;  +  12,  and 
a?2  —  5x4-6  =  0  are  equations. 

The  part  of  an  equation  on  the  left  of  the  equality  sign 
is  called  the  first  or  left  member,  that  on  the  right,  the 
second  or  right  member. 

In  an  equation  a  letter  whose  value  is  sought  is  called 
the  unknown  letter^  or  simply  the  unknown. 

The  process  of  finding  the  value  of  the  unknown  letter 
in  an  equation  is  called  solving  the  equation, 

27.  Axioms.  An  axiom  is  a  statement  whose  truth  is 
accepted  without  proof. 

In  the  solution  of  equations  constant  use  is  made  of 
four  axioms. 

Axiom  I,  If  the  same  number  is  added  to  each  member 
of  an  equation^  the  result  is  an  equation. 

Thus,  adding  5  to  each  member  of  a:  —  5  =  7  gives  x  =  12.  Axiom  I 
states  that  if  this  addition  is  performed,  the  result  is  true.  Hence 
the  original  equation  is  solved. 

Axiom  II,  If  the  same  number  is  subtracted  from  each 
member  of  an  equation,  the  result  is  an  equation. 

Thus,  subtracting  4  from  each  member  of  a:  +  4  =  10  gives  x  =  Q. 
Axiom  II  states  that  if  this  subtraction  is  performed,  the  result 
is  true. 

RE  39 


40  FIRST  COURSE  IN  ALGEBRA 

Axiom  III,  If  each  member  of  an  equation  is  multiplied 
hy  the  same  number^  the  result  is  an  equation. 

If  an  equation  is  in  the  form  -  =  6,  it  can  be  solved  by  multiplying 

each  member  by  2,  giving  x  =  12.   Axiom  III  states  that  if  this  multi- 
plication is  performed,  the  result  is  true. 

Axiom  IV,  If  each  member  of  an  equation  is  divided  by 
the  same  number  (not  zero^,  the  result  is  ayi  equation. 

If  an  equation  is  in  a  form  like  3  a:  =  12,  it  can  be  solved  by 
dividing  each  member  by  the  coefficient  of  x.  Thus,  dividing  each 
member  of  3  a:  =  12  by  3,  we  get  a;  =  4.  Axiom  IV  states  that  if  this 
division  is  performed,  the  result  is  an  equality. 

If  all  terms  containing  the  unknown  letter  are  in  one 
member  and  all  numerical  terms  in  the  other,  the  like 
terms  may  be  united  and  the  equation  solved. 

Thus  5a:  —  2a:  +  a:  =  8+15  —  3  becomes,  when  like  terms  are 
united,  4  a;  =  20,  and  dividing  each  member  by  4,  we  obtain  a;  =  5. 

Usually,  numerical  terms,  as  well  as  terms  containing 
the  unknown  letter,  will  be  found  in  each  member  of  an 
equation,  as  in  b x -{-?>  =  2 x -\-l^.  By  the  use  of  one  or 
more  of  the  preceding  axioms  it  is  always  possible  to  change 
the  form  of  such  equations  until  they  are  similar  to  the 
equation  3  a;  =  15,  which,  as  we  have  seen,  can  easily  be 
solved. 

In  changing  the  form  of  an  equation,  by  an  application 
of  the  foregoing  axioms,  it  is  important  to  note  that  we 
do  not  change  the  value  of  the  unknown.  We  merely 
discover  the  value  which  it  really  had  all  the  time.  In 
fact,  the  chief  significance  of  the  foregoing  axioms  lies  in 
their  application  to  any  kind  of  equation,  whether  it 
involves  numbers,  letters  standing  for  known  numbers 
or  for  unknown  numbers,  or  all  of  these  together. 


SIMPLE  EQUATIONS  41 

ORAL  EXERCISES 

Find  the  value  of  the  unknown  in  each  of  the  following 
equations.    State  the  axiom  or  axioms  used  in  solving  each: 

1.  2x  =  4.  15.  A -7  =  3.  24.  ?/  +  3==ll. 

2.  3;r  =  12.  x_  25.  2/  -  6  =  3. 

3.  57/,=  15.  ^^'  2~    '  ^^    X      ^ 

•^  26.  77  =  4 

4.  2^  =  16.  .«    ^      o  ^ 

17.  7:  =  3. 

5.  Ta^  =  35.  2  .„    71 


6.  .  +  1  =  2.  ^3    x^3^ 


27.  7  =  4. 

4 


7.  yt  +  3  :=  5.  *  3         •  28.  Ix  =  7. 

8.  7^  +  5  :=  11.  ^^    y^^  29.  ?  =  7, 

9.  aj  +  2  =  17.  ^ 


7 

10.  y +  9  =  16.  20    -  =  2  30.  2x  +  l  =  3. 

11.  2/-l  =  l.  '  ^        *  ^1-  2^  +  5  =  11. 

12.  ir  --  2  =  3.  21.  cc  +  4  =  7.  32.  5  ti  -  9  =  6. 

13.  ^  -  4  =  5.  22.  71  —  2  =  6.  33.  3  x  =  a^  +  4. 

14.  ic-5  =  l.  23.5^  =  30.  34.  2ic  +  l  =  2. 

EXAMPLE 

Solve  50^-8  =  2x4-19. 

Solution.  5  X-  -  8  =  2  a;  4- 19. 

Subtracting  2  x  from  each  member,  3  a;  —  8  =  19.  Ax.  II 

Adding  8  to  each  member,  3  :c  =  27.  Ax.  I 

Dividing  each  member  by  3,  a:  =  9.  Ax.  IV 

This  method  of  solution  illustrates  the  following 

Rule,  By  the  applications  of  Axioms  I  and  II  change  the 
equation  so  that  all  of  the  terms  containing  the  unknown 
number  are  in  07ie  (usually  the  left^  member  of  the  equation^ 
and  all  other  terms  in  the  other  mem.ber. 

Combine  the  terms  in  each  member. 

Divide  each  member  by  the  coefficient  of  the  unknown. 


42  FIEST  COURSE  IN  ALGEBEA 

Checking  the  solution  of  an  equation  is  often  called 
testing  or  verifying  the  result.    For  this  we  have  the 

Rule.  Substitute  the  value  of  the  unknown  obtained  from 
the  solution  in  place  of  the  letter  which  represents  the 
unknown  in  the  original  equation.  Then  simplify  the  result 
until  the  two  members  are  seen  to  be  identical. 

Check.  5x-8==2a;  +  19. 

Substituting  9  f or  :ir,  5  •  9  -  8  =  2  •  9  +  19. 
Simplifying,  45  -  8  =  18  +  19, 

or  37  =  37. 

EXERCISES 

Find  the  value  of  the  unknown  in  each  of  the  following 
equations  and  verify  results : 

1.  aj  -f  5  =  7.  13.  6  -  ^  =  9  -  4  a. 

2.  y-9  =  6.  14.  2a;  =  l-ic. 

3.  4a;-6=14.      ^  15.  5  +  3?/  =  y+4. 

4.  4cc  4-  2  =  3a?  -  4.  16.  2x  +  8  =  5aj  -1. 

5.  ^x  —  2  =  4tx  -\-  ^,  Hint.   Subtract  2x  from  each 

6.  5p  =  ^p  +  8.  member,  etc. 

7.  4cc  +  2  =  2aj.  17.  5  +  2(^  =  5^-1. 

8.  3  72,  =  w  +  3.  18.  2  -  3  71  =  71  + 1. 

9.  4a;4-2  =  ic  +  5.  19.  4x  — 2  =  7+x. 

10.  5a;  +  6  =  3cc+10.  20.  4aj  +  3-aj  +  5  =  ic-10. 

11.  4x-7  =  2cc  +  3.  21.  3  +  2^-1  =  12-3^. 

12.  3A;-5  =  7-;b.  22.  2x  +  3x  +  x=ic  +  9  +  2aj. 

23.  47/- 3 +  37/- 4  =  6y- 8. 

24.  36- 8;r  +  4-9x  =  2ic  +  10. 

25.  3x  +  12ic  +  17-cc  +  4x  =  107. 

26.  4^5 -3  =  5^-16-:- 43 A; -71. 


SIMPLE  EQUATIONS  43 

ORAL  EXERCISES 

1.  A.  rope  is  10  feet  long.  How  long  will  it  be  if  3  feet  axe 
cut  off  one  end  ? 

2.  If  3  feet  are  cut  off  one  end  of  a  rope  x  feet  long,  how 
much  remains  ? 

3.  A  river  is  d  feet  deep  at  a  ford,  and  8  feet  deeper  under 
a  bridge.    How  deep  is  it  under  the  bridge  ? 

4.  A  25-gallon  tank  is  full  of  gasoline.  If  g  gallons  are 
pumped  out,  how  many  gallons  are  left  ? 

5.  A  50-gallon  tank  is  full  of  gasoline.  If  gasoline  is 
pumped  out  until  there  are  only  x  gallons  left  in  the  tank, 
how  many  gallons  are  removed  ? 

6.  If  the  sum  of  two  numbers  is  100,  and  one  of  them  is 
riy  what  is  the  other  ? 

7.  A  pole  y  feet  long  is  cut  into  two  pieces.  If  one  piece 
is  /  feet  long,  how  long  is  the  other  ? 

8.  If  the  sum  of  two  numbers  is  s  and  one  of  them  is  10, 
what  is  the  other  ? 

9.  A  certain  chimney  is  three  times  as  high  as  a  neighbor- 
ing telegraph  pole.  If  the  pole  is  7^  feet  high,  how  high  is 
the  chimney  ? 

10.  A  rectangle  is  three  times  as  long  as  it  is  wide.  If  it  is 
X  feet  wide,  how  long  is  it  ?    What  is  its  perimeter  ? 

11.  There  are  twice  as  many  boys  in  a  certain  class  as  there 
.  are  girls.    If  there  are  n  girls,  how  many  boys  are  there  ? 

How  many  pupils  all  together  ?     How  many  people  in  the 
room,  including  the  teacher  ? 

12.  The  roof  of  a  certain  building  is  five  times  as  far  from  the 
ground  as  the  bottom  of  the  third-story  windows.  If  the  win- 
dows are  y  feet  from  the  ground,  how  high  is  the  roof  ?  How 
long  would  a  wire  have  to  be  to  reach  from  the  roof  to  the  ground 
and  back  to  a  third-story  window  ?  How  long  would  it  have  to  be 
to  reach  from  the  roof  to  the  bottom  of  the  window  direct  ? 


44  FIKST  COUKSE  IN  ALGEBRA 

13.  John's  father  is  twice  as  tall  as  John,  and  6  inches 
taller  than  John's  mother.  If  John  is  x  inches  tall,  how  tall 
is  his  father  ?    his  mother  ? 

Even  integers  are  those  exactly  divisible  by  2.  Odd  integers 
are  those  not  exactly  divisible  by  2. 

Consecutive  integers  are  integers  arranged  in  the  naturaJ 
order,  like  6,  7,  8,  9,  10,  etc. 

Consecutive  odd  integers  are  odd  integers  arranged  in  the 
natural  order,  like  5,  7,  9,  11,  13,  etc. 

Consecutive  even  integers  are  even  integers  arranged  in  the 
natural  order,  like  6,  8,  10,  12,  14,  etc. 

14.  What  is  the  difference  between  any  two  consecutive 
integers  ?  between  any  two  consecutive  odd  integers  ?  between 
any  two  consecutive  even  integers  ? 

15.  \i  n  is  an  integer,  what  is  the  next  consecutive  integer  ? 
If  X  is  an  integer,  what  are  the  next  two  consecutive  integers  ? 

16.  If  0?  is  an  odd  integer,  what  is  the  next  consecutive  odd 
integer  ?    the  next  two  consecutive  even  integers  ? 

17.  If  e  is  an  even  integer,  what  is  the  next  consecutive  even 
integer  ?   the  next  three  consecutive  odd  integers  ? 

18.  If  03  is  an  odd  integer,  is  :c  -f-1  odd  or  even  ?  \i  x  \^ 
even,  what  of  a?  + 1  ? 

19.  If  aj  is  an  odd  integer,  what  ofcc  +  3?  i«-f-6?  If?/ is 
even,  what  of?/  +  2?    ?/  +  3? 

20.  If  x  is  an  integer,  is  2  cc  odd  or  even  ?  Is  2  a^  + 1  odd 
or  even  ? 

21.  Is  the  sum  of  two  consecutive  integers  always  odd  ? 
Hint.   Express  the  integers  algebraically,  add  them,  and  divide  by  2, 

22.  Is  the  sum  of  three  consecutive  integers  always  even  ? 

23.  By  what  number  is  the  sum  of  three  consecutive  integers 
always  divisible  ? 


SIMPLE  EQUATIONS  46 

Express  the  following  statements  as  equations : 

24.  The  sum  of  5  and  x  is  8. 

25.  The  sum  of  x  and  4  is  9. 

26.  07  is  equal  to  5  diminished  by  1. 

27.  X  is  equal  to  10  diminished  by  3. 

28.  X  is  4  more  than  6. 

29.  Three  times  x  is  15. 

30.  Four  times  a  diminished  by  2  equals  18. 

31.  cc  is  6  less  than  9. 

32.  Three  times  y  is  greater  by  4  than  10. 

33.  One  half  of  x  increased  by  5  equals  9. 

34.  Two  thirds  of  x  increased  by  6  equals  14. 

35.  8  added  to  x  gives  the  same  result  as  x  taken  from  22. 

36.  2x  diminished  by  9  is  equal  to  x  increased  by  14. 

37.  If  6  is  taken  from  three  times  ic,  the  result  is  the  same 
as  when  x  is  taken  from  14. 

28.  Solution  of  problems.  In  the  solution  of  problems 
in  simple  equations  the  following  steps  are  necessary: 

1.  Head  the  problem  carefully  and  find  the  statement  which 
will  later  he  expressed  by  the  equation, 

2.  Represent  the  unknown  number  by  a  letter, 

3.  Express  the  conditions  stated  in  the  problem  as  an  equa- 
tion involving  this  letter. 

4.  Solve  the  equation. 

5.  Check  by  substituting  in  the  problem  the  value  found  for 
the  unknown. 

In  the  preceding  sentence  the  words  "  in  the  problem  " 
are  of  importance,  for  substituting  the  value  found  in  the 
equation  would  not  detect  any  errors  made  in  translating 
the  words  of  the  problem  into  the  equation. 


46  riEST  COUESE  IN  ALGEBEA 

The  foregoing  directions  for  the  solution  of  the  various 
problems  leading  to  simple  equations  are  as  definite  as 
can  be  given.  The  student  will  obtain  much  aid  from  the 
study  of  the  typical  solutions  which  occur  from  time  to 
time.  Then  one  or  more  careful  readings  of  each  problem, 
a  little  fixing  of  the  attention  upon  it,  and  an  application 
of  common  sense  will  insure  progress. 

EXAMPLE 

A  man  wishes  to  give  75  cents  to  his  two  children  so  that 
the  older  receives  15  cents  more  than  the  younger.  How  much 
shall  he  give  to  each  ? 

Solution.  The  amount  the  older  child  receives  +  the  amount  the 
younger  child  receives  =  75  cents.  Let  x  denote  the  number  of  cents 
the  younger  child  receives.  Then  x  +  lo  denotes  the  number  of  cents 
the  older  child  receives. 

By  the  conditions  of  the  problem, 

a;  +  15  +  a:  =  75. 

2  a: +  15  =75. 

2  a;  =  60. 

X  =  30. 

X  +15  =  45. 

Hence  the  younger  child  receives  30  cents,  and  the  older  receives 
45  cents. 

Check.  75  =  45  +  30,     and     45  -  30  =  15. 

PROBLEMS 

1.  One  boy  sells  5  more  newspapers  than  another.  Together 
they  sell  67  papers.    How  many  does  each  boy  sell. 

2.  It  is  desired  to  divide  a  class  of  51  students  into  two 
groups,  one  of  which  shall  contain  3  more  than  the  other. 
How  many  will  each  group  contain  ? 


SIMPLE  EQUATIONS  47 

3.  One  number  is  five  times  another,  and  their  sum  is  30. 
Find  both  numbers, 

4.  The  sum  of  two  numbers  is  72,  and  the  greater  is  three 
times  the  less.    Find  both  numbers. 

5.  One  number  is  five  times  another,  and  their  difference  is 
52.    Find  both  numbers. 

6.  A  freight  train  contains  72  cars.  It  is  desired  to 
divide  it  into  two  trains,  of  which  one  shall  contain  Jbwice 
as  many  cars  as  the  other.  How  many  cars  will  there  be  in 
each  train? 

7.  The  perimeter  of  a  certain  square  is  24  feet.  Find  the 
length  of  each  side. 

8.  A  rectangle  is  three  times  as  long  as  it  is  wide,  and  its 
perimeter  is  64  feet.    Find  its  length  and  its  width. 

9.  The  perimeter  of  a  certain  rectangle  is  78  feet.  Its 
length  is  five  times  its  breadth.    Find  its  dimensions. 

10.  A  rectangle  is  12  feet  longer  than  it  is  wide,  and  its 
perimeter  is  96  feet.    Find  its  length  and  its  width. 

11.  It  is  desired  to  cut  a  log  into  two  pieces  so  that  one 
piece  shall  be  8  feet  longer  than  the  other.  If  the  log  is  36  feet 
long,  how  long  will  each  piece  be  ? 

12.  A  stick  120  inches  long  is  to  be  cut  into  two  pieces,  one 
of  which  is  to  be  14  inches  more  than  three  times  the  length 
of  the  other.    How  long  is  each  piece  ? 

13.  The  sum  of  three  numbers  is  22.  The 'second  is  twice 
the  first,  and  the  third  is  four  times  the  second.  Find  each 
number. 

Hint.  In  solving  a  problem  involving  three  or  more  unknown  num- 
bers the  student  should  read  the  statement  carefully  and  decide  which 
of  the  numbers  to  be  found  he  should  indicate  by  x.  He  should  select 
for  this  purpose  the  number  in  terms  of  which  the  other  numbers  are- 
most  easily  expressed. 


48  FIRST  COUESE  IN  ALGEBKA 

14.  The  lirst  of  three  numbers  is  twice  the  third,  and  the 
second  is  four  times  the  third.  The  sum  of  the  three  numbers 
is  63.    Find  each  number. 

15.  It  is  desired  to  divide  a  company  of  95  soldiers  into 
three  squads  of  which  the  second  contains  5  more  men  than 
the  first,  and  the  third  contains  10  more  than  the  second.  How 
many  men  will  there  be  in  each  squad  ? 

16.  It  is  desired  to  divide  a  train  of  62  cars  into  two  sections, 
one  of  which  is  to  contain  2  more  than  twice  as  many  cars  as 
the  other.    How  many  cars  will  there  be  in  each  section  ? 

17.  Find  two  consecutive  numbers  whose  sum  is  45. 

18.  Find  two  consecutive  odd  numbers  whose  sum  is  36. 

19.  Find  three  consecutive  numbers  whose  sum  is  72. 

20.  Find  three  consecutive  even  numbers  whose  sum  is  48. 

21.  Find  five  consecutive  odd  numbers  whose  sum  is  85. 

22.  It  is  desired  to  cut  a  stick  36  inches  long  into  three 
pieces  whose  lengths  are  consecutive  integers.  How  long  will 
the  pieces  be  ? 

Biographical  Note.  John  Wallis.  Among  those  who  introduced 
and  helped  to  standardize  the  modern  algebraic  symbolism  was  John 
Wallis,  an  Englishman.  He  was  the  son  of  a  clergyman,  and,  like  most 
scholars  of  his  day,  did  not  confine  his  interest  to  any  one  subject.  He 
was  at  various  times  an  instructor  in  Latin,  Greek,  and  Hebrew,  and  for 
many  years  was  professor  of  mathematics  at  Oxford.  He  also  invented 
a  method  of  teaching  deaf  mutes  to  talk. 

During  the  wars  between  Charles  I  and  Cromwell,  Wallis's  sympa- 
thies were  with  Cromwell,  and  he  was  of  great  service  in  reading  royalist 
dispatches  writtcH  in  cipher.  In  fact,  he  was  one  of  the  most  famous 
cryptologists  of  his  day. 

Wallis  did  not  become  interested  in  mathematics  till  the  age  of  thirty- 
one,  but  devoted  himself  to  the  subject  for  the  rest  of  his  life.  One  of 
the  earliest  and  most  important  books  on  algebra  ever  written  in  English 
was  his  treatise  published  in  1685.  It  contains  a  brief  historical  sketch 
of  the  subject  which  is  unfortunately  not  entirely  accurate,  but  his 
treatment  of  the  theory  and  practice  of  arithmetic  and  algebra  has 
made  the  book  a  standard  work  for  reference  ever  since. 


JOHN  WALLIS 


CHAPTER  V 

SUBTRACTION 

29.  Subtraction  of  monomials.  The  principles  stated  on 
page  25  apply  to  the  subtraction  of  monomials  as  well-  as 
of  the  positive  and  negative  numbers  there  used.  Hence 
for  subtracting  one  monomial  from  another  we  have  the 

Rule.  Change  the  sign  of  the  subtrahend ;  then  find  the 
algebraic  sum  of  this  result  and  the  minuend. 

As  soon  as  possible  the  student  should  learn  to  change 
the  sign  of  the  subtrahend  mentally, 

EXAMPLES 

1.  From  +  8  a  take  +  5  <2. 

Solution.   +  8  a  minus  +5a  =  8a  —  5a  =  3a. 

2.  From  6  ax  take  —  2  ax. 

Solution.  6  ax  minus  —  2  ax  =  Q  ax  -{■  2  ax  =  S  ax, 

3.  Subtract  7  a%  from  -  10  a%. 

Solution.   -  10  a%  minus  7a%=^-  10  a%  -7a%  =  -17  a%. 

The  difference  of  two  dissimilar  monomials  cannot  be  written 
as  a  single  term,  but  is  expressed  by  a  binomial,  as  follows ; 

4.  Subtract  +  a  from  +  b. 
Solution.   +  i  minus  +  a  =  b  —  a. 

5.  Subtract  —  4  ^  from  3  c. 
Solution.  8  c  minus  —  4&  =  3c  +  4  6. 

6.  Subtract  4  xy  from  —  5  x^z. 
Solution.  —  5  x^z  minus  4  a:^  =  —  5  ^^^  —  4  xy. 

49 


60  FIRST  COURSE  IN  ALGEBRA 

ORAL  EXERCISES 

In  each  of  the  following,  subtract  the  second  term  from 
the  first : 

1.  8«,  3  a.  8.  4  c,  -9  c.  15.  -4  a,  -7  a. 

2.  9  a,  6  a.  9.*  -  3  a,  6  a.  16.  -  3  a,  -  8  a. 

3.  11a,  14  a.  10.  -  5  a,  8  a.  17.  -9^,  -12  a-. 

4.  7  a,  12  a.  11.  -7a;  -10a.  18.  12.T,  -5:r. 

5.  4  a,  —2  a.  12.  —  lla:^,  Gcc.  19.  —  4ac,  —  4ac. 

6.  ^a,—ba.  13.  —6  a,  —4  a.  20.  —  4ac,  4ac. 

7.  5  c,  —7  c.  14.  —^a^~^a,  21.  4ac,  —  4ac. 

Subtract  the  first  monomial  from  the  second,  and  also  the 
second  monomial  from  the  first,  in  each  of  the  following: 

22.  2x,5x,  28.   -3  c,  5  c.  34.  c,  3a'. 

23.  4  a!,  1 X.  29.    —  ac,  —  5  ac.  35.  x,  —  5y. 

24.  -2x,  -3x.  30.  8a%  -llaV.  36.  -  4  a,  6^. 

25.  -  5  a-,  -  3  a-.  31.5  a-y,  -  5  xy.  37.  -  2  a,  -5  b. 

26.  —  a-,  4  a;.  32.    —  4  ab,  0.  38.  —  2  a,  —  2  a. 

27.  —  a',  —  3  a-.  33.  a-,  y,  39.  5  a!,  —  5  a!. 

30.  Subtraction  of  polynomials.  For  tlie  subtraction  of 
polynomials  we  have  the 

Rule.  Write  the  subtrahend  under  the  minuefid  so  that 
similar  terms  are  in  the  same  column, 

Then^  changing  the  sign  of  each  term  of  the  subtrahend 
mentally^  apply  the  rule  on  page  49  to  each  column. 

Check,  hi  algebra^  as  in  arithmetic^  work  in  subtraction  is 
checked  by  use  of  the  relation 

difference  +  subtrahend  =  minuend 


SUBTRACTION  51 

EXAMPLE 

Subtract  5x  —  2 y —  1  z  -\-2  from  Sa^  +  S?/  —  5«,  and  check 
the  result. 

Solution.  3x+    ^y  —  f)z         —  minuend 

5  a:—    2?/— 72  +  2  =  subtrahend 
Check.  — 2a;H-10?/  +  22  —  2  =  difference 

Difference  +  subtrahend       3a:+    8?/  —  5;^  =  minuend 

EXERCISES 

In  Exercises  1-12,  subtract  the  first  number  from  the  second : 

1.  a  +  2,  a  +  3.       5.  2;;c-5,  5-2^.        9.  2aZ*-5c,  5c. 

2.  a  —  4,  (X  —  6.       6.   Gaj  — 3,  6  +  3aj.      10.  Sxy  —  a^dxy, 

3.  2a-3,3a-^7.     7.  9x -\- i,  4:X  -  9.      11.  0,  cc  +  4. 

4.  36t+7,  9-4a.     8.  4a,4a  — 2.  12.  2ic  — 5,0. 

In  Exercises  13-26,  subtract  the  first  polynomial  from  the 
second  and  check  the  work : 

13.  x-2y-Sz,  2x-2y-\-z. 

14.  4aj- 8?/  +  2^,  4.T- 5?/- 3^. 

15.  ?>a-2h,^a-\-h-\-2c. 

16.  5a-4Z^  +  3c,  3a-5^». 

17.  Zx^-x-h,  ^x'^-%x-2, 

18.  7i«2_4^_p-j^j^^  g^_5^2_;j^3^ 

19.  a  —  X  -\-  y,  h  —  X  —  y. 

20.  3a  —  2^+c  +  6,  4(x  —  Z>-|-5c. 

21.  2a  +  2^/-2c-4,  a-3/>. 

22.  2a-3:r-f  4c,  4x-3a-3c-ll. 

23.  a  -\-  h  —  c,  c  —  a  —  h. 

24.  2  a^  -  3  ah  -f-  /A  3  //  -  7  a^  +  8  ah. 


52  FIEST  COTJESE  IN  ALGEBEA 

25.  3  a?^  —  3  x^y  —  3  xy^,  4  x^y  —  5  xy^  —  y\ 

26.  a^-c^-Sa'c  +  S  ac",  5  a^c  +  4  c^  +  3  ^^  +  2  ac\ 

In  Exercises  27-35,  find  the  expression  which  added  to  the 
first  will 'give  the  second  : 

27.  x  —  2y-\-z,2x-\-5y  —  Sz.       30.  a  —  h  +  2c,5. 

28.  Tec— 9?^  — 3^,  5a;  — 2^/  — 4^.      31.  ax^  +  bx  +  c,  2bx  —  c, 
29    5  a  —  4  ^  +  6  c,  6  a  —  3  5.  32.  3  a5  +  c,  3  a;y  —  ^. 

33.  2x  —  4.y  —  z,  0. 

34.  a'  -  2aG  ■{-3c%2c^-3ac-{-4.  a\ 

35.  2  x'^y  —  3  ici/^,  ?/^a;  +  yx'^  —  z. 

In  Exercises  36-42,  find  the  expression  which  subtracted 
from  the  first  will  give  the  second : 

36.  x  —  2y  —  z^?>x  —  2y  —  z. 

37.  cc^  -  7  ic  - 10,  14  cc  -  8  +  3  x'', 

38.  X  —  y  ~  z,  5x-\-3y  —  %z. 

39.  4-8xV'  +  5ic-6. 

40.  a^  +  ^V  +  c*,  2  c^  -  3  aV  -  4  a*. 

41.  3a  +  bh-c,0, 

42.  4a  +  6^^- 8?/,  a-  L2. 

43.  Subtract  the  sum  of  a^  -\-2ah  —  l  and  a^ -{-12  ah  —  20 
from  a"  +  13  ah-  30. 

44.  Subtract  the  sum  of  a  —  3 Z>  +  c  and  4a  +  5^  —  6c-{-4 
from  a  —  h  -\-  c  —  X. 

45.  From  the  sum  of  5  cc  +  3  x^y  — 15  xy'^  and  —  6  a?  — 12  x}^ 
+  7?/^:;^  subtract  11  a?  —  bx^y  -f  7  2/^x. 

46.  From    the    sum    of    4  ahc^  —  3  a^^c  +  2  a^^c   and    6  ahe^ 
-  5  a^^c  -  4  a^^^c  subtract  2  ^^^c^  -  3  a%G  +  7  aPc. 

47.  From  the  sum  of  Sx  —  4:xy  —  2z  and  7a;y  —  4^  —  3a; 
take  the  sum  oi  5z  —  2xy  —  a%c  and  9x  —  6 a^hc  —  z. 


SUBTEACTION  63 

Find  the  algebraic  sum  of : 

48.  (4x~32/-f-6)  +  (3x  +  52/-l®). 

49.  (7  c -f  5  cZ  -  e)  -  (4  c  +  5  (^  -  9  e). 

50.  {x^  +  2x  +  5)-{-(2x'-{-x-10)-(x^-5x-h  3). 

51.  (x-\-3y-2z)  +  (4:X-5y-h3z)-(3x-2y-6z), 

52.  (5x  +  Sy  --  z)  +  (4:y  +  7 z)-(x  -  y  -{-  Sz). 

53.  Ax-Sy-\-7 -(2x-oy-4.)-{'(4.x-S), 

54.  3G-5d-e-(5G-\-6d  +  lle)-(5c  +  Ae). 

55.  3a  +  3^>-4e-(-35-3c-4)-(4a  +  a5-8c). 

56.  x«-  c^  +  3caj2  _  3c2x  -.(x»  +  c^-2c''x  -  ^.cx"),     ■ 


CHAPTER  VI 

IDENTITIES  AND  EQUATIONS  OF  CONDITION 

31.  Kinds  of  equations.  Equations  are  of  two  kinds: 
identities  and  equations  of  condition. 

An  identity  is  an  equation  in  which,  if  the  indicated 
operations  are  performed,  the  two  members  become  pre- 
cisely ahke,  term  for  term. 

4  •  6 
Thus,  4«5  +  3'4  =  8-5 — -  is  an  identity,  for,  if  we  perform 

the  indicated  operations,  it  becomes  20  +  12  =  40  —  8,  or  32  =  32. 

Similarly,  2a  +  36  —  4  =  3a--2&  +  (o6  —  a  —  4)  is  an  identity, 
for,  if  we  perform  the  indicated  addition  in  the  second  member,  the 
equation  becomes  2a  +  36  —  4  =  2a  +  36  —  4,  in  which  the  two 
members  are  alike,  term  for  term. 

An  identity  which  involves  letters  is  true  for  any  nu- 
merical values  of  the  letters  in  it. 

Thus  the  literal  identity  (a  +  3)2  =  a2  4.g^^9  becomes,  when 
a  =  5,  (5  +  3)2  =  52  +  6  .  5  +  9,  or  82  =  25  +  30  +  9,  or  64  =  64.  If 
a  is  zero,  the  identity  becomes  (0  +  3)2  =  0  +  6  •  0  +  9,  or  9  =  9. 

An  equation  in  one  unknown  which  is  true  only  for 
certain  values  of  the  unknown  is  an  equation  of  condition, 
which  for  brevity  we  shall  usually  refer  to  simply  as  an 
equation. 

The  statement  4  a:  =  a;  +  12  is  true  only  when  a:  =  4.  If  4  is  sub- 
stituted for  X,  the  equation  becomes  the  identity  4  •  4  =  4  +  12,  or 
16  =  16.  Clearly  the  statement  is  false  if  0,  or  3,  or  any  value 
other  than  4  is  put  for  x\  it  is  true  on  condition  that  x  be  4,  and 
on  no  other. 

54 


IDENTITIES  AND  EQUATIONS  OF  CONDITION     55 

Similarly,  x^— 6x+Q  =  0is  true  when  x  =  2  or  when  a?  =  3.  In 
che  first  case  x^  — 6x  +  6  =  0  becomes  2^  —  5  •  2  +  6  =  0,  or  4  —  10 
+  6  =  0,  or  0  =  0.  In  the  second  case  we  obtain  3^  —  5  •  3  +  6  =  0, 
OT  9  — 15  +  6  =  0,  or  0  =  0.  Plainly  the  statement  obtained  is  false 
when  —  2  is  put  for  x^  for  then  it  becomes  (—  2)^  —  5(—  2)  +  6  =  0, 
or  4  +  10  +  6  =  0,  or  20  =  0.  Similarly,  any  value  other  than  2  or  3, 
when  put  for  x,  gives  a  relation  between  numbers  which  is  not  true. 

Every  equation  of  condition  may  be  regarded  as  asking  a  question. 
Thus  the  equation  3  a;  +  2  =  15  asks,  "What  number  when  multi- 
plied by  3  and  the  product  increased  by  2  gives  15  as  the  result?  " 

The  equations  used  in  solving  the  problems  on  pages 
46-48  are  equations  of  condition.  The  conditions  there 
expressed  in  ordinary  language  in  the  problems  were  trans- 
lated into  the  algebraic  language  of  equations. 

Instead  of  the  equality  sign,  the  sign  ~  (read  is  identical 
with,  or  is  identically  equal  to)  is  sometimes  used  for  em- 
phasis if  the  expi^ession  is  an  identity. 

Thus  3  a  =  2  a  +  a  may  be  written  3  a  =  2  a  +  a. 

32.  Root  of  an  equation.  A  number  or  literal  expression 
which  being  substituted  for  the  unknown  letter  in  an 
equation  reduces  it  to  an  identity  is  said  to  satisfy  the 
equation. 

Thus  it  has  been  shown  that  4  satisfies  the  equation  ^x  =  x  -\- 12, 
and,  similarly,  the  literal  expression  3  a  satisfies  the  equation  x  —  6 
=  3  a  -  5. 

A  number  or  number  symbol  is  called  a  root  of  an  equation 

if  it  satisfies  the  equation. 

The  process  of  checking  the  solution  is  really  finding 
out  whether  the  result  obtained  is  a  root  of  the  equation 
or  not.  It  should  be  particularly  noted  that  after  the  root 
has  been  substituted  in  place  of  the  unknown,  the  equation 
of  condition  becomes  an  id^fttity. 


66  FIRST  COUESE  IN  ALGEBRA 

ORAL  EXERCISES 

In  Exercises  1-10,  state  which  expressions  are  identities 
and  which  are  equations  of  condition :  , 

1.  3  +  7  =  13-3.  e.  2x  =  4:X-3x-{-x. 

2.  2a;  =  8,  7.  x  +  7  =  4.-\-x  +  S. 

3.  Sa  —  5  =  2a.  8.  Sx -l  +  x +1  -  4.x  =  0. 

4.  a;  +  l  =  l+aj.  9.  x  +  4.-Sx-\-2  =  6x-S. 

5.  a;-2  =  2i»-a  10.  3x -3  =  4.x  -  2 -x-1. 

In  Exercises  11-20,  select  from  the  numbers  at  the  right 
of  each  equation  those  which  satisfy  that  equation : 

11.  4x-12  =  0.  1,2,3. 

12.  2aj  =  ir  +  2.  1,2,4. 

13.  x-\-3  =  4:X-6.  0,  -1,3. 

14.  3aj-4  =  2x  +  l.  -2,  1,  5. 

15.  ic  +  6  =  3  a;  - 10.  8,  5,  -  4. 

16.  a;  +  1  —  2  a;  +  3  =  0.  7,  8,  -  3. 

17.  0^2  _|.  3^  4.  2  =  0.  -1,  1,  0. 

18.  x^-16  =  6x.  8,  -2,  3. 

19.  4a;2_|_4^^3  2,  J,  3. 

20.  5^2  =  3x.  1,  2,  0. 

Is  the  number  at  the  right  of  each  of  the  following  equations 
a  root  of  that  equation  ? 

21.  3a;  -  24  =  0.  8.  25.  a;^  -  16  =  4a;  +  5.  7. 

22.  4a;-15  =  x.  3.  26.  3a;  +  7  =  5a;-l.  4. 

23.  5a; -12  =  2a;.  2.  27.  f  +  5y  +  4.  =  0.  2. 

24.  a;2  +  6  a;  -  3  =  0.  4.  28.  ^  +  3  =  2  ^  -  1.  4. 

33.  Transposition.  In  solving  the  equation  5a;— 4  =  18 
we  add  4  to  each  member.  If  we  indicate  this  addition, 
the  equation  becomes 

5;?:-4  +  4  =  18+4. 


IDENTITIES  AND  EQUATIONS  OF  CONDITION     57 

Now  in  the  first  member  —4  +  4  =  0,  so  that  these  two 
numbers  may  be  omitted,  and  the  equation  becomes 

5a;  =  18+4. 

Upon  comparing  this  with  the  original  equation,  it  is  seen 
that  —  4  has  vanished  from  the  first  member  of  the  original 
equation,  and  +  4  has  appeared  in  the  second  member  of 
the  new  equation. 

Again,  if  we  subtract  ?>y  from  both  members  of  the 
equation  6y=3y  +  12, 

we  get  the  equation      6^  —  3^=12, 

which  differs  from  the  original  equation  only  in  having 
—  3  ?/  in  the  first  member  instead  of  +  3  ^  in  the  second. 

It  thus  appears  that  a  term  may  he  omitted  from  one 
member  of  an  equation^  provided  the  same  term  with  its  sign 
changed  from  -\-  to  —  or  from  —  to  -\-  is  written  in  the  other 
member.    This  process  is  called  transposition. 

Hereafter,  in  order  to  simplify  an  equation,  instead  of  sub- 
tracting a  number  from  both  members  or  adding  a  number 
to  both  members,  as  illustrated  in  the  example  on  page  41, 
the  student  should  use  transposition,  as  it  is  usually  more 
rapid  and  convenient.  He  should,  however,  always  remem- 
ber that  the  transposition  of  a  term  is  really  the  subtraction 
of  that  term  from  each  member  of  the  equation. 

Like  terms  in  the  same  member  of  an  equation  should 
be  combined  before  transposing  any  term. 

Note.  Our  word  algebra  is  derived  from  the  Arabic  word  for 
transposition.  The  process  by  which  one  passes  from  the  equation 
px  —  q  =  x^  to  the  equation  px  =  x^  +  q  was  known  as  al-jehr.  This 
is  the  first  word  in  the  title  of  an  Arabic  book  on  algebra  which  was 
translated  into  Latin.  For  some  reason  only  this  part  of  the  title 
remained,  and  by  the  early  part  of  the  seventeenth  century  al-jehr,  or 
algebra,  was  the  common  name  given  to  the  whole  subject. 


68  riRST  COURSE  IK  ALOEBRA 

ORAL  EXERCISES 

State  the  term  or  terms  which  must  be  added  to'  both 
members  of  the  following  equations  in  order  to  transpose  the 
underscored  terms.  What  does  each  equation  become  after 
transposition  ? 

1.  x^—5  =  4..  7.  2ic  +  3  +  3a5  +  5  =  4^-h6. 

2.  aj  +  12  =  4.  S.  5x-2-Sx-it-6  =  7~x. 

3.  cc  -  3  =  2.  9.  x^  +  4:x  =-  3. 

4.  x''-{-6x  =  5x-S.  10.  3x-2=-x\ 

5.  x^—4:X  =  5.  11.  5x^— 4:X  =  Sx  —  l. 

6.  2x^-\-3x  +  l  =  2x-4..      12.  8x  -  2x^=  4  -  4^^. 

EXAMPLE 

Solve  the  equation  5  x  —  7  -i-  2  x  -{-  6  =  11  x  ~  5  —  6  x  -\- 12, 

Solution.  5a:-7+2a:  +  6=lla:-5-6a:  +  12. 

Combining  like  terms,  7a;  —  l  =  5a;  +  7. 

Transposing,  7a:  —  5a:  =  7  +  l. 

Combining  like  terms,  2  a:  =  8. 

Dividing  by  2,  a;  =  4. 

Check.  5:zr-7+2a:  +  6=lla;-5-6a;  +  12. 

Substituting  4  for  a:,  20  -  7  +  8  +  6  =  44  -  5  -  24  +  12. 
Combining,  27  =  27. 

EXERCISES 

Solve  the  following  equations,  and  check  results : 

1.  4:X-l  =  Sx-}-3.  6.  4tk  +  l-2k  =  4., 

2.  4:x  +  5  =  x-10,  7.  16  +  4ic-4  =  x  +  12. 

3.  -6?/- 2  =  1- 92/.  8.  4ic-15  =  15-0^-5. 

4.  6x  +  3-2x  =  2T,  9.  5n -\-ll  -  2n=  6  -  4:^ 

5.  42/  +  3  =  2  +  5?/.  10.  2a^- 1  =  29  + 7a;. 


IDEI^TITIES  AND  EQUATIONS  OF  CONDITION     59 

11.  Sk-{-9  +  5k-SS  =  0.  15.  4a; -}- 3  =  15  +  2a; -f  6. 

12.  3  A  -  20  +  8  A  -  24  =  0.  16.  3  y  +  5  +  y  +  3  =  0. 

13.  6a;-13  +  2a;  +  3  =  0.  17.  9  -  8 A  +  2  =  3  -  4 A. 

14.  a;  +  2-5a;  =  -9x  +  12.  IS,  3  -  5x  +  2  =  5 +7x. 

19.  8a;  — 3a;  =  15a; +  4 -13a;. 

20.  8  +  73/-13  =  2/-27-5y. 

21.  3a;-15-10a;-9+16a;-21  =  0. 

22.  18  +  5a;- 6-2a; +l  +  3a;-25  =  0. 

23.  0  =  18-4a;  +  27  +  9a;-3-fl6a;. 

24.  5n-^^=4.n-j-5  =  7n  —  S-2n  +  5. 


ORAL  EXERCISES 

Represent  a  niiiiiiber : 

1.  Greater  by  4»feihan  x.  6.  Four  greater  than  a  -\-b. 

2.  Greater  by  a  than  x.  7.  a;  greater  than  a  -{-  b. 

3.  Less  by  5  than  n.  8.  Seven  less  than  a;  —  2y. 

4.  Less  by  b  than  n.  9.  a  less  than  2c  —  b. 

5.  Three  times  ?/.  10.  c  less  than  three  times  x, 

11.  a  greater  than  four  times  n. 

12.  Three  less  than  five  times  y. 

13.  Eight  greater  than  three  times  n. 

14.  One  part  of  10  is  4.   What  is  the  other  part? 

15.  One  part  of  x  is  5.   What  is  the  other  part? 

16.  One  part  of  8  is  y.  What  is  the  other  part? 

17.  One  part  of  n  is  p.  What  is  the  other  part? 

18.  One  part  of  x  is  a.  What  is  the  other  part? 

19.  One  part  oi  x  -\-  y  i^  z.  What  is  the  other  part? 


60  FIRST  COUESE  IN  ALGEBRA 

20.  The  sum  of  two  numbers  is  12.    If  one  of  them  is  5. 
what  is  the  other? 

21.  The  sum  of  two  numbers  is  12.    If  one  of  them  is  p 
what  is  the  other? 

22.  The  difference  of  two  numbers  is  8.    If  the  greater  is  25, 
what  is  the  other  ? 

23.  The  difference  of  two  numbers  is  8.    If  the  greater  is  x, 
what  is  the  other? 

24.  The  sum  of  two  numbers  is  25.    If  one  of  them  is  a. 
what  is  the  other? 

25.  The  sum  of  two  numbers  is  s.    If  one  of  them  is  9, 
what  is  the  other? 

26.  The  sum  of  two  numbers  is  a.    If  one  of  them  is  x 
what  is  the  other? 

27.  The  difference  of  two  numbers  is,  6.    The  less  number 
is  4.   What  is  the  other? 

28.  The  difference  of  two  numbers  is  d.    If  the  less  of  them 
is  4,  what  is  the  other? 

29.  The  difference  of  two  numbers  is  d.    If  the  less  of  them 
is  n,  what  is  the  other? 

30.  What  is  the  excess  of  19  over  6?  9  over  ti? 

31.  By  how  much  does  24  exceed  16?  24  exceed  x?  y  ex- 
ceed 8?  a  +  ^»  exceed  12? 

32.  How  much  greater  is  45  than  18  ?    than  33  ?   than  a  ? 
How  much  greater  is  x  than  y  ? 

33.  How  much  less  is  15  than  32?   than  a?   How  much 
less  is  x  than  y  ? 

34.  By  how  much  does  a  +  6  exceed  a?    a  +  6  exceed  h  ? 
4  ic  —  3  exceed  3  cc  ? 

34.  Translation    of    problems    into    equations.     In    the 

solution    of    problems    the    writing    of    the    equation    is 
nothing  more  than  translating  from  ordinary  speech  int® 


IDENTITIES  AND  EQUATIONS  OF  CONDITION     61 

the   language    of   algebra.     Sometimes   it   is   possible   to 
translate  the   statement   of  the  problem,  word   by  word, 
into  algebraic  symbols. 
For  example, 

Four  times  a  certain  number,  diminished  by  6, 
4         X  n  -  6 

gives  the  same  result  as  the  number  increased  by  30. 
=  n  +  30 


PROBLEMS 

1.  What  number  increased  by  16  is  equal  to  37  ? 
Hint.  n  +  16  =  37 

2.  What  number  diminished  by  23  is  equal  to  9  ? 

3.  To  what  number  must  28  be  added  so  that  the  result 
may  be  15  ? 

4.  From  what  number  must  15  be  subtracted  so  that  the 
result  may  be  24  ? 

5.  If  a  certain  number  is  doubled  and  the  result  dimin- 
ished by  13,  the  remainder  is  49.    What  is  the  number  ? 

6.  If  a  certain  number  is  doubled  and  the  result  increased 
by  16,  the  sum  is  12.    What  is  the  number  ? 

7.  If  a  certain  number  is  trebled  and  the  result  diminished 
by  16,  the  remainder  is  equal  to  the  original  number.  What  is 
the  number  ? 

8.  Three  times  a  certain  number,  less  23,  equals  twice  the 
number,  less  8.    Find  the  number. 

9.  Five  times  a  certain  number,  increased  by  12,  equals 
eight  times  the  number,  diminished  by  6.    Find  the  number. 

10.  Four  times  a  certain  number,  less  7,  equals  twice  the 
number,  plus  21.    Find  the  number. 

11.  What  number  is  as  much  less  than  90  as  it  is  greater 
than  16  ? 


62  FIRST  COUESE  IN  ALGEBEA 

12.  What  number  exceeds  16  by  twice  as  much  as  52  exceeds 
the  number  ? 

13.  A  certain  number  added  to  14  gives  the  same  result  as 
that  obtained  when  this  number  is  subtracted  from  58.  What 
is  the  number  ? 

14.  If  9  is  added  to  twice  a  certain  number,  and  12  is 
subtracted  from  five  times  the  number,  the  results  are  the  same. 
Find  the  number. 

15.  Five  times  a  certain  number,  plus  16,  equals  three  times 
the  number,  plus  10.    What  is  the  number  ? 

16.  The  sum  of  two  numbers  is  62,  and  their  difference 
is  10.    Find  the  numbers. 

Solution.    The  greater  number  +  the  less  number  =  62. 

There  are  two  unknowns  in  this  problem,  but  both  can  be  repre- 
sented in  terms  of  the  same  letter,  thus : 

Let  n  =  the  less  number. 

Then  n  +  10  =  the  greater  number,  since  the  smaller 

is  10  less  than  the  greater. 

Placing  these  symbols  in  the  principal  statement,  we  have 
n  + 10  +  n  =  62. 

Combining,  2  w  +  10  =  62. 

Transposing,  2  n  =  62  —  10. 

Combining,  2  n  =  52. 

Dividing  by  2,  n  =  26,  the  less  number, 

and  n  +  10  =  36,  the  greater  number. 

Check.  36  +  26  =  62,  36  -  26  =  10. 

17.  The  sum  of  two  numbers  is  58,  and  their  difference 
is  12.    Find  the  numbers. 

18.  The  sum  of  two  numbers  is  8,  and  their  difference  is  42. 
Find  the  numbers. 

19.  The  sum  of  two  numbers  is  21,  and  the  second  is  5 
greater  than  the  first.    Find  the  numbers. 


IDENTITIES  AND  EQUATIONS  OF  CONDITION    63 

20.  The  sum  of  two  numbers  is  85,  and  one  exceeds  the 
other  by  41,    Find  the  numbers. 

21.  The  sum  of  three  numbers  is  55.  The  second  is  5 
greater  than  the  first,  and  the  third  is  17  greater  than  the  first. 
Find  the  numbers. 

22.  The  sum  of  three  numbers  is  16.  The  second  is  10  less 
than  the  first,  and  the  third  is  2  less  than  twice  the  first.  What 
are  the  numbers  ? 

23.  Find  three  consecutive  integers  whose  sum  is  90. 

24.  Find  three  consecutive  odd  integers  whose  sum  is  33. 

25.  A  rectangle  whose  perimeter  is  32  feet  is  three  times  as 
long  as  it  is  wide.    Find  its  dimensions. 

26.  A  rectangle  whose  perimeter  is  102  feet  is  twice  as  long 
as  it  is  wide.    Find  its  dimensions. 

27.  The  length  of  a  rectangle  is  7  feet  more  than  twice  the 
width.    Its  perimeter  is  110  feet.    What  are  its  dimensions  ? 

28.  A  lawyer  can  afford  to  spend  only  $40  per  week  for 
ofiS.ce  help.  He  must  pay  the  office  boy  |5 ;  and  he  estimates 
that  the  stenographer's  work  is  worth  two  thirds  as  much  as 
the  bookkeeper's.    How  much  should  he  pay  each  ? 

29.  Mr.  Brown  can  allow  his  three  children  all  together  a 
dollar  and  a  half  per  week  for  spending  money.  In  order  to 
pay  his  car  fare  to  school  John  needs  50  cents  per  week  more 
than  Elizabeth,  while  James  requires  only  half  as  much  as 
Elizabeth.    What  allowance  will  each  child  receive  ? 

30.  A  farmer  offers  his  son  a  reward  if  he  will  shingle  the 
roof  of  the  barn  in  6  working  days.  The  roof  contains  120 
courses  of  shingles.  The  son  wishes  to  plan  the  work  so  that 
he  may  lay  each  day  two  courses  less  than  the  previous  day. 
How  many  courses  must  he  lay  the  first  day  ? 


CHAPTER  VII 

PARENTHESES 

35.  Removal  of  parentheses.  In  solving  exercises  and 
problems  it  is  often  necessary  to  inclose  several  terms  in 
a  parenthesis.  Sometimes  it  is  necessary  to  inclose  this 
parenthesis  with  other  terms  in  a  second  parenthesis  or 
even  in  a  third.  To  avoid  confusing  the  different  paren- 
theses, brackets  [  ]  and  braces  {  }  are  also  used. 

The  parenthesis,  the  brackets,  and  the  braces  are  called 
signs  of  aggregation.  For  convenience,  brackets  and  braces 
are  often  spoken  of  as  parentheses. 

Occasionally  the  vinculum,  or  bar,  is  used  in  the  same 
way  as  a  parenthesis.    Thus  —  m  +  2  a  =  —  (m  +  2  a). 

In  the  solution  of  equations  and  in  other  algebraic  work 
it  is  frequently  necessary  to  remove  all  signs  of  aggrega- 
tion. This  removal,  while  it  depends  only  on  the  princi- 
ples of  addition  and  subtraction  which  have  already  been 
learned,  requires,  nevertheless,  some  special  study  to  acquire 
speed  and  accuracy. 

The  value  of  12 +(5  — 3)  is  the  same  as  that  of  12 
+  5  —  3,  or  14.     Similarly,  a-{-(b  —  c')=a-{-b  —  c. 

The  plus  signs  preceding  the  parentheses  in  12  +  (5  —  3)  and 
a  +  (6  —  c)  belong  to  these  parentheses  respectively  and  vanish 
with  them,  whereas  the  plus  signs  understood  before  5  and  h 
within  the  parentheses  are  supplied  when  we  write  12  +  (5  —  3) 
=  12  +  5  —  3  and  a  +  (/;  —  c)  =  a  +  ^  —  c.  In  the  expression  12 
4-  (—  5  —  3)  the  sign  of  5  must  be  retained,  and  we  have  12  +  (—  5  —  3) 
=  12  -  5  -  3  =  4. 

64 


PARENTHESES  65 

Therefore  we  have  the 

Principle.  A  parenthesis  and  the  sign  hefore  it^  if  plus, 
may  he  removed  from  an  expression  without  changing  the 
signs  of  the  terms  which  were  inclosed  by  the  parenthesis. 

In  the  expression  12  — (5  — 3)  the  sign  before  the 
binomial  shows  that  (5  —  3)  is  to  be  subtracted  from  12. 
Hence  we  change  the  signs  of  the  terms  in  the  subtrahend 
and  find  the  sum  of  the  resulting  terms  and  the  minuend. 

Thus  12  -  (5  -  3)  =  12  -  5  +  3  =  10.  This  is  obviously  correct, 
for  12  -  (5  -  3)  =  12  -  2  =  10. 

Similarly,  a  —  (h  —  c)  becomes  a  —  h  -\-  c  when  the  signs  of  the 
subtrahend,  (6  —  c),  are  changed  and  the  result  is  added  to  a. 

The  minus  signs  preceding  the  parentheses  in  12  —  (5  —  3)  and 
o  —  (6  —  c)  vanish  with  the  parentheses,  and  the  plus  signs  under- 
stood before  5  and  h  within  the  parentheses  are  changed  when  we 
write  12  -  (5  -  3)  =  12  -  5  +  3  and  a  -  (&  -  c)  =  a  -  6  +  c. 

Therefore  we  have  the 

Principle,  A  parenthesis  and  the  sign  hefore  it,  if  minus, 
may  he  removed  frorn  an  expression,  provided  the  sign  of  each 
term  which  was  inclosed  hy  the  parenthesis  he  changed. 

These  principles  may  also  be  applied  to  remove  the 
parentheses  used  to  inclose  the  numbers  in  Chapter  II. 

When  one  parenthesis  incloses  another,  either  the  outer 
or  the  inner  parenthesis  may  be  removed  first.  It  is  best 
for  the  beginner  to  use  the 

Rule,  Rewrite  the  expression,  omitting  the  innermost  paren- 
thesis, changing  the  signs  of  the  terms  which  it  inclosed  if 
the  sign  preceding  it  he  minus  and  leaving  them  unchanged 
if  it  he  plus, 

Comhine  like  terms  that  may  occur  within  the  new  inner- 
most par&nthesis. 

Repeat  these  processes  until  all  the  parentheses  are  removed. 


66  FIEST  COUESE  m  ALGEBRA 

EXAMPLE 

Eemove  the  parentheses  from  8  —  (3  —  2  a)  +  (4  —  5  a). 
Solution.    8-(3-2a)4-(4-5a)  =  8-3  +  2a  +  4-5a  =  9-3a. 

The  student  should  observe  that  the  removal  of  a  paren- 
thesis is  merely  the  performance  of  an  indicated  addition 
or  subtraction.  Thus  3  —  2  a  in  the  example  above  is  a 
subtrahend  which  is  to  be  taken  from  8  +  4  —  5  a  even 
though  it  comes  before  4  —  5  <x. 

The  student  should  exercise  great  care  in  the  removal 
of  a  parenthesis  preceded  by  a  minus  sign.  Errors  in  this 
connection  are  common  and  they  persist  long. 

ORAL  EXERCISES 

Eead  the  following  expressions  after  removing  the  paren- 
theses : 

1.  8+(4  +  2).  5.  x+(^  +  ^).  9.  a-(a-h). 

2.  8 +(4 -2).  Q.  x+Qj-z).  l(^.  a-\-{-a-\-h), 

3.  8 -(4 +  2).  7.  x-{y^-z).  11.  x  +  ^j +{x  -  y), 

4.  8 -(4 -2).  8.  x-{y-z).  12.  a-h-ip-a), 

13.  a-^-{^-a-\-x).  17.  -  (3a  -  c)-(2c  -  5  a). 

14.  {a-c)-(G^a).  18.  x -{2  a  -  Zx  -  2c). 

15.  -{2a-h)  +  {2h-a),  19.  {- x  +  a)-{a  -  ^x), 

16.  -{a-^c)^-{2-x).  20.  _  (- a +  cc) +  (- a- 5ir). 

EXERCISES 

Remove  the  parentheses  and  combine  like  terms : 

1.  14  _  (6  -  3)  -  5.  3.  (7  -  5  +  2)  -  (6  -  4)  -f  12. 

2.  10+(7-4)-(9-7).    4.  lla-{4.a-^a)^-{ioa-ay 

5.  {2h-ba)-(4.a-h--la). 

6.  a-{h-a)-\-(2h-^c). 

7.  a-h-{G-^d)'\-{a-h)-(b^G), 


PARENTHESES  67 

8.  (x-y)-(2y-^x)^{x-4.y), 

9.  x-{x-y-2z)-{^z-\-y  +  ^)^{x-^), 
10.  7  -  [8  -  (3  -  10)]  -  (13  -  25). 

Hint.    7  -  [8  -  3  +  10]  -  13  +  25,  etc. 

Observe  that  we  may  first  remove  the  inner  parenthesis,  —  (3  —  10), 
and  also  remove  the  last  parentheses  at  the  same  time.  Doing  this  often 
saves  rewriting  and  decreases  the  probability  of  error. 

11.  8-[5-(4-6)]-(6-15). 

12.  12-(8-17)-[(4-l),-8].  . 

13.  16-[3-(2-5)]-}-(3-5)-(8-4). 
^14.  a+[2a-(3a-2^)]  +  (3^-2a). 

lb.  a-{-[pa-{^x-2ay\-(4.a-Zx), 
'   16.  2a-\Jda-{px-^a):^^-{^a-lx). 

17.  How  many  parentheses  may  be  removed  the  first  time 
Exercise  18  is  rewritten?  Exercise  19?  20?  21?  22?  23? 
24?  25? 

,18.  (^x  -  Q,y)-\_-  4.x  -(4.Z  -  y)-2z'\, 
\ll9.  [3.'r-(2  2/  +  ^)]-[-(3y-2aj>+5^]. 

20.  [(a  +  3)-(x-5>]-[a  +  3+(i^-5)]. 

21.  7-[-6-{-4+(9-10)>  +  ll]. 

22.  5  a  -  [2  c&  +  (-  3  a  +  4  hy-  (a  -  8  hy-\-  4  a]. 

.  23.  2cc-3y-[{3^-7a;-(?/-4^)--9a;}.  +  «]. 

24.  {4a-[26i-(8«^  +  2/^>+4]— (4Z^-6)}. 

25.  _5aj  +  [15aj-{llic-(2a;-7^-4)--34-22]. 

26.  (4y-7cc)-{3aj-[4i^-(7  2/-4ic)-(-5?/  +  3cc)]}. 

Sometimes  it  is  necessary  to  remove  some  of  the  signs  of 
aggregation  in  an  expression,  leaving  others.  In  the  following 
remove  the  parentheses,  retaining  the  brackets,  and  simplify 
the  results  as  much  as  possible : 

27.  [(a  +  &)+c],  [(«  +  ^)-4 

28.  [4aj+(3^-52/)],  [4x-(3^-5y)]. 


68  FIRST  COURSE  IN  ALGEBRA 

29.  l(a  -h)  +  {h-2  a)],  l(a  -b)-(b-2  a)]. 

30.  [(3a-2b)  +  (2b-3a)'],  [(3  a  -  2b)- (2b  -  30)]. 

31.  [(a-2b)  +  (3c-d)l  [(a-2b)-(3c-d)-]. 

32.  [(4^ -3) +  (5 2/ -7)],  [(4^ -3) -(5  ^-7)].        .   • 

33.  [(x'  -  a')  +  (f-2  a')l  [(x'  -  a')  -(f-2  a')}, 

36.  Inclosing  terms  in  parentheses.    Obviously, 

16  +  9  -  5  -16  +  (9  -  5),  for  each  equals  20. 
Similarly,  a  ■+■  b  —  c  =  a  -\-  (b  —  c). 

That  the  two  preceding  expressions  are  equal  may  be 
seen  by  removing  the  parentheses  according  to  the  first 
Principle  on  page  65. 

These  processes  are  illustrations  of  the 

Principle,  One  or  more  terms  may  he  inclosed  in  a  paren- 
thesis preceded  hy  a  plus  sign^  without  changing  the  sign  of 
any  of  the  terms. 

The  expression 

17  +  8  -  3  =  17  -  (-  8  +  3),  for  each  equals  22. 
Similarly,  a  -\-h  —  c  =  a  —  (—  5  +  ^)> 

and  a  —  h  -{-  c  =  a  —  (h  —  c'). 

That  the  right  member  in  each  of  these  eases  is  aaiother 
form  of  the  left  may  be  seen  by  removing  the  parentheses 
aecording  to  the  second  Principle  on  page  65. 

These  processes  are  illustrations  of  the 

Principle,  One  or  more  terms  may  he  inclosed  in  a  paren- 
thesis preceded  hy  a  minus  sign,  provided  the  sign  of  each 
term  thus  inclosed  is  changed. 


PAEENTHESES  69 

EXERCISES 

In  the  following,  inclose  in  a  parenthesis  preceded  by  a  plus 
sign  all  the  terms  containing  the  letters  x  or  ?/,  and  inclose  in 
a  parenthesis  preceded  by  a  minus  sign  all  the  other  terms: 

\,  x'-a^-^a.  3.  3/2  -  9  ^>2  +  6  aZ»  -  a''. 

2.  12a  +  «2-9-4al  4.  10aZ;+x2-a2-25Z>l 

5.  ^2  _  ^2  _  4^2  _|_  ^2  _  ^2  _  2xy. 

6.  -  ^ah  -\-  x^  -  ^IJ"  ^  if  -  o?  -  2xy, 

7.  16 :r2  -  a^  -  16a^y  -  4  +  4  a  +  4 ?/2. 

.  8.  a;2  -  Z'^  _  j^Q  ^^  j^Yiah-  36  a^  +  25  if. 
9.  2aZ>  -  a^  H-  a;2  +  2:i;y  -  Z.2  ^  /. 
10.  x^  -1^  xij  -  a^  ^\%  a  ^-  64.2f  -  64. 

11.  In  Exercises  1-4,  inclose  the  last  two  terms  in  a  paren- 
thesis preceded  by  a  negative  sign. 

12.  In  Exercises  5-10,  inclose  the  last  three  terms  in  a 
parenthesis  preceded  by  a  negative  sign. 


CPIAPTER  VTTT 

MULTIPLICATION 

37.  Product  of  terms  containing  unlike  letters.  Tlie  stu- 
dent is  familiar  with  the  fact  that  the  factors  of  a  product 
may  be  written  in  any  order. 

For  example,  2.4  =  4.2. 

Similarly,  a  ^  h  —  h  »  a. 

This  principle  is  called  the  Commutative  Law  of  Mul- 
tiplication. 

Further,  2  a^  x  3  =  2  x  8  x  a^  ^  6  a^^ 

and  2  a  X  ^y/j  =  2  X  '^  X  a  X  b  =  Q  ab. 

Similarly,  G  a:^  •  5  ?/  =  6  •  5  •  2-  •  ?/^  =  30  x'^y'^. 

Also,  4  a/> .  a  ^2  =  4  .  3  .  ab  •  z^  =  12  abz\ 

Up  to  the  present  we  have  assumed  that  the  various 
operations  of  multiplication  in  any  product  may  be  per- 
formed in  any  order. 

That  is,  (3.  2) 4  =-3 (2.  4)  =24.  In  general  terms 
a(b  '  c)  —  (a'  h^c.  This  merely  tells  us  that  a  multiplied 
by  the  product  of  b  and  c  is  the  same  as  the  product  of  a 
and  b  multiplied  by  e.  This  principle  is  called  the  Asso- 
ciative Law  of  Multiplication. 

Biographical  Note.  Sir  William  Rowan  Hamilton.  It  is  strange 
that  of  all  the  topics  treated  in  this  book  the  last  to  be  thoroughly  under- 
stood by  mathematicians  are  those  appearing  in  the  first  chapters.  But 
in  all  the  sciences  it  is  often  most  difficult  to  answer  the  questions  that 
at  lirst  sight  seem  quite  obvious.  Any  child  can  ask  what  electricity  is, 
but  the  wisest  scientist  cannot  tell.    He  can  only  exi)lain  what  electricity 

70 


SIK  WILLIAM  ROWAN  HAMILTON 


MULTIPLICATION  71 

does.  It  is  easy  to  ask  how  the  earth  came  to  be  revolving  around  the 
sun  with  the  moon  revolving  around  it,  but  even  the  deepest  students  of 
astronomy  differ  in  their  theories  of  how  it  came  to  be.  And  so  in  mathe- 
matics, long  after  many  of  the  more  complicated  processes  of  algebra 
were  completely  understood,  the  simple  laws  of  operation  of  numbers 
were  surrounded  with  haze.  One  of  the  men  who  did  most  to  clarify  the 
nature  of  these  laws  was  Sir  William  Kowan  Hamilton  (1805-1865).  He 
was  born  in  Dublin,  Ireland,  where  he  lived  most  of  his  life.  He  was  a 
precocious  boy,  and  at  the  age  of  twt|^e  was  familiar  with  thirteen  lan- 
guages. He  devised  kinds  of  numbers  that  do  not  follow  the  same  laws 
as  those  that  we  use  in  algebra,  and  so  threw  a  flood  of  light  on  the  nature 
and  properties  of  these  common  numbers.  He  was  the  first  to  recognize 
the  importance  of  the  Associative  Law,  and  called  it  by  that  name.  Most 
of  his  works  are  very  advanced  in  character  and  are  difficult  to  read. 

38.  Product  of  terms  containing  like  letters.  By  the  defi- 
nition of  an  exponent  (p.  9),  a^  =  a  -  a^  and  (j?=a  *  a  -  a. 

Therefore  d^  Y.  a^  =  a-  a  x  a-  a  -  a  =  w"  =  oP--^^. 

Similarly,  I  x  h^  x  b^  =  h  x  h-h-h  xb'h'h'b'b  =  h^=b^+^  +  ^. 

In  like  manner  3^  x  3*  x  3^  =  3  •  3  x  3  •  3  •  3  •  3  x  3  •  3  •  3  •  3  •  3  =  S^i 

^3-2  +  4  +  5. 

Also,  ay'^  x  y^  =  ay^  ■=  ay"^  +  ^, 

and  2abxna^  =  6a%  =  Q  a^  +  ^~b, 

and  4  x^yz  x  6  xy^  =  20  x^y^z  =  20  a:2  +  ^y^  +  H. 

Therefore  we  have  the 

Principle.  The  exponent  of  any  letter  in  the  product  is 
equal  to  the  sum  of  the  exponents  of  that  letter  in  the  factors. 

This  is  expressed  in  general  terms,  tlius : 

The  law  of  signs  for  the  multiplication  of  positive  and 
negative  numbers,  given  on  page  27,  applies  to  literal  terms 
as  well. 

Thus  +  2  d^  X  (+  3  a^)  =  +  6  a\ 

+  2  a2  X  (-  3  a5)  =  -  6  al 
-2a2  X  (+^a^)  =  -.Qa\ 
-2a2  X  (-3a5)=  +  Q  a\ 


72  MEST  COURSE  IK  ALGEBRA 

For  tlie  multiplication  of  two  monomials  we  have  the 
Rule.  Obeying  the  7'ule  of  signs  for  multiplication^  write  the 
product  of  the  numerical  coefficients  followed  by  all  the  letters 
that  occur  in  the  multiplier  and  the  multiplicand^  each  letter 
having  as  its  exponent  the  sum  of  the  exponents  of  that  lette/ 
in  the  multiplier  ayid  the  multiplicand, 

ORAL  EXERCISES 

Perform  the  following  indicated  nuiltiplications : 


1.  (3)  (-6). 

6. 

x"-:^.                11.  x'^.x'. 

2.  (-2) (7). 

7. 

x^'-xK                12.   -x'2.(;»^). 

3.  (-8)  (-3). 

8. 

x}  .  x'^                13.   -  a^^ .  (-  x^). 

4.   (-4a^)(5). 

9. 

x^'x".                14.   -  a;'' .  (- 3  aj«). 

5.  3 2^.  5x.                   1 

LO. 

cr'^..Tl                 15.   -x''-(2a^). 

16.   (-2ar)(3.T^). 

21.   (-7a^)(-2a^^). 

17.   (3a^')(-  2x^). 

22.   {-\-2a^){-?,a'x). 

18.   {^x^){-^x:'). 

23.   (-4a^i^)(-j-2^^). 

19.    (-3a;«)(-4.r^). 

24.   (-5aV.)(-7ai«). 

20.   {-(jx'')(Ao'). 

25.   (+ 3  aV)  (4  «;^i»2). 

26.    (-4:^^)'^                30. 

(- 

-9a)(-10).         34.   (-lla;)(3a;). 

27.   (7)  (-5  a).          31. 

(- 

-^axf.                35.   (7ic)(-3a-). 

28.   (3  a)  (-8).          32. 

(4 

:a)(-5./).          36.   {-a)\ 

29.   {-2yy.                33. 

{(bahc)\                    37.    {-2af. 

38.   (3a2a.-)'l 

47.   (-3a2af. 

39.  (-2  a)  (-4^2). 

48.   (3^^)(-v/). 

40.   (5a^)(8a«). 

49.  {^xSj){-2x^). 

41.   {-5xf. 

50.   (- 6xV)'- 

42.   (a«)(-16a). 

51.   {-xS/){~ah,% 

43.   (-4a^)(-6«.^). 

52.   (2«ir2)«. 

44.   (+3./)^. 

53.   (5a^)(-4a2)(-3a). 

45.   {4.x){6y){^xy). 

54.   {-2a^){^ay, 

46.  {2»x){-2yY, 

55 .   (3  ax)  (-  2  a^o^)  C-  7  ^.r^y 

MULTIPLICATION  73 

39.  Multiplication  of  a  polynomial  by  a  monomiaL  Clearly 
2  (5  +  8)  is  equivalent  to  2  •  5  -f-  2  .  3,  each  expression  being 
equal  to  16. 

Similarly,  a {h -f  c)  =  ah-]-  ac.  This  principle  is  called  the 
Distributive  Law  of  Multiplication. 

Therefore,  for  the  multiplication  of  a  polynomial  by  a 
monomial,  we  have  the 

Rule.  Multiply  each  term  of  the  polynomial  hy  the  monomial 
and  write  in  succession  the  resulthig  terms  with  their  proper 
signs. 

Example.       o  x-  —  2  xy  +  4  //  —  5  <i  —  6 

2xy 

Product,     Qxhj  —  ^  x'^if  +  '6xif  —  \^  axy  —  12xy 

EXERCISES 

Multiply : 

1.  X  -\-2  ])y  2.  .  S.  5x"-  —  2x  —  4:  by  a;l 

2.  X  —  4:hj  X.  9.  —  4 x'^  +  6 a;  —  5  by  6x^. 

3.  ic"  +  5  by  2x.  10.  2^^  _  3^  _  2  by  -  ^x", 

4.  3x^  +  4  by  2ic.  11.  x^- 3x^-4  by  -5x^ 

5.  x'^-2xbyx2.  12.  2x  -  Sx^  -  2x^  by- 2x«. 

6.  ?/2  —  3  7/  +  2  by  3  ?/.  13.  7  xy  —  x  +  2/  by  3  xy. 

7.  y3_5^2_^3y  _j^  ]^3^y2^    14^  ^2_  2^y  +  y2  ^y  -3X7/ 

15.  a^  -  a%^  +  ^'  by  -  a%\ 

16.  -  aV  -2ax-{-7Phj  -  i  abx, 

17.  7x«- 8x2H-12x- 6by -4x^. 

18.  -  9  a"-  12  ax  +  42  x^  by  3  axl 

Perform  the  multiplication  indicated : 

19.  7(2x-3).  22.   -9(-4r^  +  ^^). 

20.  5x(x-2/).  23.   -3x(2x-7). 
Z\,   -8(3x-7).  24.  6x'^(9x3-4x). 


74  FIRST  COURSE  IN  ALGEBRA 

25.  -^(x'-2x-  6).  28.   (x^  -  ax -\-  a^)  (-  2  a'x). 

26.  bxij{x^—Q^x-{-  9).  29.   —1  ah  {ax^  -  hx -{-  c). 

27.  -  3  X  {ax  -bx-S  ex"),      30.  4  x2(-  3  x  +  7  x^-~  x% 

40.  Multiplication  of  polynomials.  Clearly  (5  +  3)(7— 4) 
=  8  •  3  =  24.  The  multiplication  may  also  be  performed 
as  follows:  (5  +  3)  (7  -  4)=  5  (7- 4) -f  3  (7  -  4)  =  35 
-20  4-21-12  =  24. 

Similarly,  (2  2:  +  3)  (4 :?;  -  5)  =  2  rz;  (4  a;  ~  5)  +  3  (4  ^  -  5) 
=  8^:2-10:^  +  12:1^-15,  or  "ix^^lx-lh. 

In  general  terms  {a  -\- V)  {c  -^  d)  —  a  (c  ■\- d?) -\- h  (c  ■\-  c?) 
=  ac  -\-  ad  -{-he  -\-  hd. 

This  gives  for  the  multiplication  of  polynomials  the 

Rule,  Multiply  the  multiplicand  by  each  term  of  the  multi- 
plier in  turn,  and  add  the  partial  products. 

Example.  3  x  —  2 

2x  +3 


Multiplying  by  2  a:,        6  x'^  —  4  x         =  first  partial  product. 
Multiplying  by +  3,  +  9  x  —  Q  =  second  partial  product. 

Complete  product,  Qx^-\-6x  —  6  =  sum  of  partial  products. 

EXERCISES 

Multiply : 

1.  ::c  +  4  by  :r  +  3.  9.  3  a:  —  2  by  2  a:  -  3. 

2.  2 ic  +  3  by  a?  +  3.  10,  -  3x -{-11a  hj  5  x  -  a. 

3.  4  cc  +  7  by  3  cc  +  2.  11,  ax  —  bx  hj  ex  -\-  dx. 
^,  3  X  —  5  hj  3  X  -\-  S.  12,  —  ex  -\-  d  hj  bx  —  cx\ 

5.  3a:-2by  2cc  +  3.  13.  4.x  -  3y^  hj  6x -{- 5]/. 

6.  6  -  4  a  by  5  a  -  7.  14.  cc^  _  5  ^  +  6  by  x  -  3. 

7.  2x-{-yhj  x-^3y.  15.  3^^  _  3^  _  7  ^^^  2^  +  4. 

8.  2a:  —  3 y  by  3  x  —  2  y.  16.  x^  —  x^  +  f  hj  x -}- y. 


MULTIPLICATION  76 

17.  aV  —  2  a^x  -i-  4:a^  hj  ax  -\-2  a. 
IS.  Sx^^x^-5xhj  2x^-5x\ 

19.  2x^-7x  +  12hyx^-3x-^5. 

20.  x^'-2x-3hy  x^-2x-S. 

Expand : 

21.  (3x^-5x-l)(2x--3).  24.  (3y--t/-2y. 

22.  (x^  -  3x  -  2)(x''  ^  2x  +  3).  25.  (4.x-2x^-5y. 

23.  (t^-3t  +  2y.  26.  (3x^-4.x  +  7y. 

41.  Powers.  A  power  of  a  number  is  the  product  ob- 
tained by  using  the  number  as  a  factor  one  or  more  times. 

For  example,  8,  or  2^,  is  the  third  power  of  2 ;  81,  or  3*,  is  the 
fourth  power  of  3  ;  and  32  x^,  or  (2  xy,  is  the  fifth  power  of  2  x. 

42.  Arrangement.  A  polynomial  is  said  to  be  arranged 
according  to  the  descending  powers  of  a  certain  letter  wh^n 
the  exponents  of  that  letter  in  successive  terms  decrease 
from  left  to  right.  Thus  2x'^— 5x^— 6x  +  8  is  arranged 
according  to  the  descending  powers  of  x.  Again,  4  —  2  y 
+  y2  and  x^—  3 x^y  +  3 xy^  —  y^  are  arranged  according  to 
the  ascending  powers  of  y. 

Whenever  it  is  possible  to  arrange  the  multiplier  and 
the  multiplicand  in  the  same  order  with  respect  to  the 
same  letter  it  should  be  done,  as  the  addition  of  the 
partial  products  is  then  much  more  easily  performed. 

43.  Degree.  The  degree  of  a  term  with  respect  to  any 
letter  which  does  not  appear  in  the  denominator  is  deter- 
mined by  the  exponent  of  that  letter  in  the  term. 

Thus  X,  3  xy,  and  4  a^xz  are  of  the  first  degree  in  a:,  and  3  xy^  is  of 
the  second  degree  in  y. 


76  FIRST  COURSE  IN  ALGEBRA 

The  degree  of  a  term  with  respect  to  two  or  more  letters 
which  do  not  appear  iii  the  denominator  is  determined  by 
the  8um  of  the  exponents  of  those  letters  m  that  term. 

Thus  5  x^y  is  of  the  fourth  degree  in  x  and  y ;  4  a%cVy  is  of  the 
sixth  degree  in  a,  ft,  and  c. 

44.  Check  of  multiplication.  The  work  of  multiplication 
can  be  checked  by  giving  a  small,  convenient  numerical 
value  to  each  letter  involved  and  finding  the  correspond- 
ing numerical  values  of  the  multiplier,  the  multiplicand, 
and  the  product.  The  product  of  the  numerical  values  of 
the  multiplier  and  the  multiplicand  should  equal  the 
numerical  value  of  the  product. 

The  least  positive  integer  which  gives  a  reliable  check 
by  the  method  outlined  above  is  the  number  2.  This  is 
true  only  when  one  letter  is  involved.  If  more  than  one 
letter  is  involved,  the  check  is  not  certahi  if  2  is  substi- 
tuted for  each  letter. 

The  number  1  is  very  convenient  to  use  in  checking, 
but  it  will  not  check  exponents,  since  x^  =  a^  =  a^  =  x^^, 
etc.,  if  a:=l. 

EXAMPLE 

Multiply  3x^—5-\-x^—2xhyx^—6  —  5x,  and  check  result : 

Solution  and  check.  Arranging  both  niultipUer  and  multiplicand 
in  descending  powers  of  x  and  multiplying,  we  obtain : 

3x''-^x^-2x-r)        =24  +  4-4-5=:         10    (If  a;  =  2) 

x^-ryx-iy =_4_10-G       :=-    12 

8a;S+         x^-      2x^-    5.r2  -228 

-    lox^-      5a:3  + 10x2  + 25a: 

-   18a:3-    6a;2  +  12y  +  30 

Product,      3a:5-    Ux*-    25a;3-      x^  +  37x  +  30 

Product,    96     -224     -200     -   4     +74+30    =-228    (If  a:  =  2) 

Since  —  228  is  obtained  in  both  steps  of  the  check,  the  result 
is  correct. 


MULTIPLICATION  77 

ORAL  EXERCISES 

1.  Arrange  3  a;^  —  8  ic^  —  7  —  a?*  —  2  a?  in  descending  order. 
K  2.   Arrange  a^  -f-  /y'*  +  6  aVj'^  —  4  aJ/  —  4  a^b  in  descending  order 
with  reference  to  the  letter  b ;  with  reference  to  the  letter  a. 

3.  Why  should  multiplier  and  multiplicand  be  arranged  in 
the  same  order  before  attempting  to  multiply  ? 

4.  What  is  the  value  of  a%^  and  of  a%^  ifa  =  h  =  2? 

5.  What  point  in  checking  does  the  preceding  question 
bring  out  ? 

6.  If  1  or  0  is  correctly  substituted  for  the  letter  or  letters 
in  a  check  of  multiplication  and  the  two  results  do  not  agree, 
is  the  multiplication  necessarily  incorrect  ? 

7.  Why  is  a  check  of  multiplication  unreliable  when  the 
number  1  is  substituted  for  the  letter  or  letters  involved  ? 

1^  8.  If  one  interchanges  the  multiplier  and  multiplicand  and 
performs  the  multiplication  a  second  time,  would  this  give  a 
check  on  the  first  ?    Explain. 

EXERCISES 
Multiply,  and  check  every  fifth  exercise : 

1.  3x'-5x-2hj2x-3, 

2.  3x^-i-2x  -  7  by  2 ^^^^  5^  _  3 

3.  x^  -f  2  X  _  4  by  x'  +  2ic  +  4. 

4.  3a2-8a-l  by  a^+2rt-3. 

5.  2a^2_7^.  _2  by  itself. 

6.  2ir.«-3a;  +  2  by  itself. 

7.  (r  —  I  a  -f  I  by  a^  —  a  -\-  1. 

8.  x^  —  xij  -{■  if  by  x^  +  xy  -f  y\ 

9.  3x'3  +  5a;2  -  i»  +  2  by  ic^  -  2ic  4- 3. 

10.  a  -  3  -  «« -  2  aM)y  2  a  -  3  a2  -  1. 

Hint.   Arrange  multipHer  and  multiplicand  in  descending  order  first 

11.  3a-  a^  -^6hy  Aa-3a--5. 


78  riEST  COUKSE  IK  ALGEBEA 

Expand : 

L12.  (x^-x-5)(Sx^-2x-4.). 

13.  (3x  -  x^  +  x"-  6)(5  -  x^  -  Sx). 

14.  I3x  -  2  a  -(2  a  -  Sx)XSx  -  2  a  +(2a  -  Sx)']. 

15.  (4.a-5a^  +  7-ha^)(S  +  a^-a  +  a'). 

.    16.  (5x  -  3  +  Sx^)(S  -  5x^+  2x^  -  9x). 

17.  (x^ -  2xy  +  3y^)(x^  +  2xy  +  3y''). 

18.  (x^y  —  2/^cc)  (4  cci/  —  5  x^y)  (3  cc^?/  —  2  xy^). 

19.  (a:;^  +  y^  +  ^^  —  iry  —  33;$;  —  ?/^) (x  +  ?/  +  ^). 

20.  (a  +  5  +  c)l  23.  (2a  -  4&  +  3)1 

21.  (2a-3^  +  4c)l  24.  («^  -  2Z»  +  3c  -  4c^)l 

22.  (c  +  cZ  -  1)1  25.  (^  +  y  +  ^y- 

Expand  and  collect  like  terms  in : 

26.  (x  +  2yy-(x-2tjy.  31.  {4:X  -  32jy -(2  x  +  dyf. 

27.  (x-3y-\-(3-xy.  32.  (x  -  3y-(2x -ly. 

28.  (i:c  +  y)«  +  (a^  -  tjy.  33 .  (2  x^  -  3  «)  (2  0^2  ^  3  a)l 

29.  (2x-3ay-(3a-2xy.      34.  (a;8_5>^(^8^  4)_(^6_^  ^2^^^ 

30.  (ax-5a7jy-(ax-mjy,      35.  [a  +  (a2+3)][a-(ci2+3)]. 

36.  [a  -  (x'  _  3)]  [a  +  (x^  -  3)]. 

37.  [x-a-rix"-  2)][(x  -  a)  -  (x^  -  2)]. 

38.  (2aj-3)(3^.'-^-5a^)-(4^2_3^>^(2aj-7)-(3aj-6)2. 


CHAPTER  IX 

PARENTHESES  IN  EQUATIONS 

45.  Simple  equations  involving  parentheses.  In  handling 
parentheses  it  is  very  easy  to  acquire  careless  habits,  which 
are  difficult  to  overcome.  Accuracy  in  such  work  demands 
especial  care  in  removing  each  parenthesis  that  is  preceded 
by  a  minus  sign. 

EXAMPLE 

Solve  the  equation  3(2x -\-l)  —  (4.x —  7)  =  16. 


Solution.                      3  ( 

:2  ^  + 1)  - 

(4  a:- 

-7)  = 

=  16. 

Removing  parentheses. 

,     Qx  +  d 

-4:X 

'+7: 

=  16. 

Combining, 

2x- 

flO  = 

=  16. 

Transposing, 

2x-. 

=  6. 

Dividing  by  2, 

X  - 

=  3. 

Check.                      3  (2  • 

3  +  1)  -  (4  .  3 

-7)  = 

=  16. 

Simplifying, 

21 

-5  -- 

=  16, 

or 

16  = 

=  16. 

EXERCISES 

Solve  and  check  : 

1.  2(;r-f-3)  =  12.  8.  16  +  2(4  7/-7)-12y=0. 

2.  5(cc-l)=30.  9.  5a;-4(4-rr)-ll  =  0. 

3.  4(;r  +  6)  =  16.  10.  2(aj4-l)-3  =  3(x-l). 

4.  2(3-^^)4-1  =  2.  11.  4(2aj-5)  +  15  =  3(a;+10> 

5.  5(aj-3)  +  14  =  4.  12.  3(a^  +  6)+8  =  5(6+x). 

6.  5(x-7)4-8aj  =  4.  13.  7(x -f  5)=  4(x  +  8)-f  3. 

7.  4(4aj-l)  +  3=;r.  14.  7(y  -  2)- 2(3  +  2/)=  0. 

79 


80  FIKST  COURSE  IN  ALGEBKA 

15.  97/ -3(2  7/ -4)=  2(5-7/)  f  7. 

16.  2n~9(2n-\-4.)=2(n-9). 
'17.  7x-12-2(x- 5)=x-U. 

18.  5(3A-l)-7A  =  3(7^4-7)-l. 

19.  (n  -  4)  (71  4-  8)  =  7  -^3  -  n)  (n  -f-  5). 
Solution.  Expanding,        (?i'^  +  4:  n  —  82)  ==  7  —  (15  —  2  7i  —  ;r). 
Removing  [)arentheses,       ii"  +  4  li  —  o2  =  7  —  15  +  2  n  +  ii"^. 
Subtracting  ii^  from  each  member,  and  combining, 

4  n  -  82  =  -  8  4-  2  n. 

Transposing  and  combining,  2  ii  =  24. 

Dividing  by  2,  ?/,  =  12. 

Check.  (12  -  4)  (12  +  8)  ==  7  -  (8  - 12)  (12  +  5). 

Simplifying,  8  •  20  =:  7  -  (- 0  •  17)  ; 

that  is,  160  =7  —  (—  153), 

or  160  =  160. 

20.  (x  +  3)-  -  (x  -I-  5f  :=  -  40. 

21.  (x  +  2)^-  -  (x  -  4)'^  +  48  =  0. 

22.  (2x  -  4)(3.T  -  6)=  6.x-  +  72. 

23.  (.r  -f-  4)  (a^  -h  6)  =  (.r  +  18)  (a-  +  13). 

24.  (A  +  2)  (k  +  3)  =  (A  -  5)  (k  ~  2). 

25.  (/^  -  7)(5  +  /^-(/c  -  r))(/.'  +  7)  +  5  =  0. 

ORAL  EXERCISES 

1.  The  length  of  a  rectangle  is  ;:t^  —  3  and  its  width  is  4. 
What  is  its  area?  its  perimeter? 

2.  The  length  of  a  rectangle  is  a  and  its  breadth  is  b. 
What  is  its  area?  its  perimeter? 

3.  Express  as  a  product  the  area  of  a  rectangle  whose  length 
is  3  a?  +  2  and  whose  breadth  is  cc  —  1.  What  is  its  perimeter  ? 

4.  Each  of  three  sheep  cost  $40.  What  was  the  cost  of  all? 

5.  Each  of  n  horses  cost  $200.    What  represents  the  cost 
of  all? 


PARENTHESES  IN  EQUATIONS  81 

6.  Each  of  a  books  cost  b  cents.  What  represents  the  cost 
of  all? 

7.  What  is  the  total  cost  of  x  hats  at  a  dollars  each,  and 
7j  hats  at  b  dollars  each? 

8.  What  is  the  cost  of  h  horses  at  a?  -f  15  dollars  each? 

9.  Represent  the  total  cost  of  a  chairs  at  2/4-4  dollars 
each,  and  b  chairs  at  ^s  —  2  dollars  each. 

10.  A  is  n  years  old.  What  will  be  his  age  6  years  hence? 
X  years  hence?  What  was  his  age  5  years  ago?  a  years  ago? 

11.  A's  age  in  years  is  three  times  B's.  If  B  is  a?  years  old, 
represent  A's  age  (a)  now,  Qi)  7  years  hence,  (c)  4  years  ago, 
(d)  the  sum  of  their  ages  4  years  hence. 

12.  A's  age  is  2n  —  3  years.  What  will  be  his  age  12  years 
from  now?  b  years  from  now?  What  was  his  age  7  years  ago? 
X  years  ago  ? 

13.  A  and  B  each  have  d  dollars.  If  A  gives  B  five  dollars, 
how  much  will  each  then  have  ? 

14.  A  and  B  each  have  x  -\- 25  dollars.  If  B  gives  d  dollars 
to  A,  how  much  will  each  then  have? 

15.  If  A  has  X  +  25  dollars,  and  B  has  2  ic  -f  8  dollars,  express 
as  an  equation  each  of  the  following  statements : 

(a)  A  has  as  many  dollars  as  B. 

(b)  A  and  B  together  have  $250. 

(c)  A  has  $15  less  than  B. 

(d)  If  A  gains  $75  and  B  loses  $20,  they  have  equal  amounts. 

16.  If  A's  age  is  n  years,  B's  3  ri  -[-  5  years,  and  C's  2  ti  —  4 
years,  express 

(a)  the  ages  of  A,  B,  and  C  six  years  hence ; 

(b)  the  ages  of  A,  B,  and  C  five  years  ago. 

Express  each  of  the  follo.wing  statements  as  an  equation  : 

( c )  The  sum  of  the  ages  of  A  and  B  six  years  hence  will 
be  50  years. 


82  riKST  COUESE  IK  ALGEBEA 

(d)  The  difference  of  the  ages  of  C  and  A  four  years  ago 
was  19  years. 

(  6 )  In  12  years  A  will  be  as  old  as  B  is  now. 

(/)  Six  years  ago  C  was  as  old  as  A  will  be  15  years  hence. 

{g)  In  X  years  B  will  be  54  years. 

{h)  In  five  years  the  sum  of  the  ages  of  A,  B,  and  C  will 
be  100  years. 

17.  A  picture  is  10  inches  wide  and  12  inches  long  and  has 
a  frame  2  inches  wide.  What  are  the  outside  dimensions  of 
the  frame? 

18.  If  the  frame  in  the  preceding  exercise  were  x  inches 
wide,  what  would  represent  the  outside  dimensions  of  the 
frame?  the  area  of  the  picture  and  frame?  the  area  of  the 
picture  ?  the  area  of  the  frame  ? 

46.  Problems  involving  parentheses.  The  following  prob- 
lems involve  two  or  more  unknowns  and  the  use  of  paren- 
theses. One  of  the  unknowns  can  always  be  represented 
by  a  single  letter  and  the  others  by  binomials  involving 
this  letter  and  one  or  more  numbers.  It  will  be  necessary 
in  some  of  the  problems  to  inclose  each  of  these  binomials 
in  a  parenthesis  and  to  think  of  them  and  use  them  as  if 
each  represented  a  single  number.  When  the  student  can 
use  a  binomial  in  this  way  as  readily  as  he  uses  a  smgle 
letter,  like  x^  he  has  made  considerable  progress  in  the 
algebraic  way  of  thinking. 

So  far  as  possible  the  method  of  translating  the  problem 
into  the  symbolic  language  of  algebra  which  was  mentioned 
on  page  61  should  be  followed. 

For  example :  The  sum  of  two  numbers  is  34.  Four  times  the 
less  equals  three  times  the  greater,  plus  10.    Find  the  numbers. 

Here  there  are  two  unknowns,  the  greater  number  and  the  less. 
Each  can  be  represented  in  terms  of  a  letter,  or  this  letter  and  a 
number,  as  follows: 


PAEENTHESES  IK  EQUATIONS  83 

Let  n  represent  the  less  number.  Then  34  —  n  must  represent 
the  greater. 

By  the  conditions  of  the  problem, 
four  times  the  less  equals  three  times  the  greater,  plus  10. 
4         X  n        =  3  X  (34-n)    +    10, 

or  4  n  =  3  (34  -  n)  +  10. 

Again :  Seven  times  A's  age  two  years  ago  equals  five  times  his 
7  X  («-2)  =        5        X 

age  ten  years  hence. 

(a  +  10), 
or  7(a-2)  =  5(a+10). 

PROBLEMS 

1.  The  sum  of  two  numbers  is  49.  Twice  the  greater 
equals  7  plus  five  times  the  less.    Find  each  number. 

2.  The  sum  of  two  numbers  is  45.  Ten  times  the  less  equals 
300  minus  five  times  the  greater.    Eind  each  number. 

3.  The  sum  of  two  numbers  is  15.  Twice  one  of  them, 
minus  ten  times  tlie  other,  equals  zero.  What  are  the  numbers  ? 

4.  Separate  75  into  two  parts  such  that  29  plus  four  times 
the  less  equals  three  times  the  greater. 

5.  Twice  a  certain  integer,  plus  four  times  the  next  con- 
secutive integer,  is  106.    What  are  the  integers  ? 

6.  Three  times  a  certain  integer,  subtracted  from  four  times 
the  next  consecutive  integer,  is  17.    What  are  the  integers  ? 

7.  The  sum  of  two  numbers  is  15.  Seven  times  one  number 
equals  54  plus  ten  times  the  other.    Eind  the  numbers. 

8.  Twice  a  certain  number  equals  240  plus  five  times  a 
second  number.  The  sum  of  the  numbers  is  15.  Find  the 
numbers. 

9.  The  perimeter  of  a  rectangle  is  90  feet,  and  five  times 
the  greater  side  plus  four  times  the  less  equals  207.  What 
are  the  sides? 


84  FIEST  COURSE  IN  ALGEBEA 

10.  The  difference  of  the  squares  of  two  consecutive  integers 
is  35.    Find  the  integers. 

11.  The  difference  of  the  squares  of  two  consecutive  integers 
is  71,    Find  the  integers. 

12.  The  difference  of  the  squares  of  two  consecutive  odd 
integers  is  104.    Find  the  integers. 

13.  The  square  of  an  integer  plus  the  square  of  the  next 
consecutive  integer  is  17  less  than  twice  the  square  of  the 
greater  integer.    Find  the  integers. 

14.  The  difference  of  the  squares  of  two  consecutive  odd 
integers  is  48.    Find  the  integers. 

15.  The  product  of  two  consecutive  odd  integers  is  42  less 
than  the  square  of  the  greater  integer.    Find  the  integers. 

16.  The  product  of  two  consecutive  even  integers  equals 
44  increased  by  the  square  of  the  smaller.    Find  the  integers. 

17.  A's  age  in  years  is  three  times  B's,  and  C  is  10  years 
older  than  B.  The  sum  of  their  ages  is  45  years.  Find  the 
age  of  each. 

18.  A's  age  in  years  is  twice  B's,  and  C  is  7  years  older 
than  A.  Six  years  hence  the  sum  of  their  ages  will  be  85 
years.    How  old  is  each  ? 

19.  A  is  10  years  older  than  B,  and  C  is  6  years  younger 
than  B.  Six  years  ago  the  sum  of  their  ages  was  40  years. 
Find  the  age  of  each. 

20.  A  is  2  years  more  than  twice  as  old  as  B,  and  C  is 
7  years  younger  than  A.  In  6  years  the  sum  of  their  ages 
will  be  75  years.    Find  the  age  of  each. 

21.  A  is  now  50  and  B  is  36  years  old.  How  many  years 
ago  was  A  three  times  as  old  as  B  ? 

22.  A  is  now  19  years  old  and  B  is  54  In  how  many  years 
will  A  be  exactly  half  as  old  as  B  ? 


PAEENTHESES  IN  EQUATIONS  85 

23.  A  is  30  years  older  than  B.  In  20  years  A  will  be  twice 
as  old  as  B.    Find  the  age  of  each  now. 

24.  A  is  three  times  as  old  as  B.  In  15  years  A  will  be 
twice  as  old  as  B.    Find  the  present  age  of  each. 

25.  A's  age  is  8  years  more  than  twice  B's  age.  Sixteen  years 
ago  A  was  four  times  as  old  as  B.   Find  the  age  of  each  now. 

26.  A  square  has  the  same  area  as  a  rectangle  whose  length 
is  8  inches  greater,  and  whose  breadth  is  4  inches  less,  than  the 
side  of  the  square.    Find  the  dimensions  of  each. 

Solution.   By  the  conditions  of  the  problem, 

the  area  of  the  square  =  the  area  of  the  rectangle. 
Let  s  =  the  length  of  the  side  of  the  square  in  inches. 

Then        5  +  8  =  the  length  of  the  rectangle  in  inches, 
and  5  —  4  =  the  breadth  of  the  rectangle  in  inches. 

Now  the  area  of  the  square  is  s  •  s,  or  s^,  square  inches,  and  the 
area  of  the  rectangle  is  (s  +  8)  (s  —  4),  or  s^  +  4  s  —  32,  square  inches. 
Therefore  s^  =  s^  +  4:S  -  32.  (1) 

Solving  (1),        s  =  8,  the  length  of  the  side  of  the  square ; 
s  +  8  =  16,  the  length  of  the  rectangle, 
and  5  —  4  =  4,  the  breadth  of  the  rectangle. 

Check.  The  area  of  the  square  is  8  •  8  =  64  square  inches,  and  the 
area  of  the  rectangle  is  also  16  •  4  =  64  square  inches. 

27.  A  square  field  has  the  same  area  as  a  rectangular  field 
whose-  length  is  15  rods  greater,  and  whose  breadth  is  10  rods 
less,  than  the  side  of  the  square.  Find  the  dimensions  of  each 
field. 

28.  A  tennis  court,  for  singles,  is  3  feet  shorter  than  three 
times  its  breadth.  The  distance  around  the  court  is  210  feet. 
Find  the  length  and  the  breadth  of  the  court. 

29.  A  tennis  court,  for  doubles,  is  6  feet  longer  than  twice 
its  breadth.  The  perimeter  of  the  court  is  228  feet.  Find  the 
dimensions  of  th^  court. 


86  FIRST  COUESE  IN  ALGEBEA 

30.  The  breadth  of  a  basket-ball  court  is  20  feet  less  than 
its  length.  The  perimeter  of  the  court  is  80  yards.  Find  the 
dimensions. 

31.  The  perimeter  of  a  rectangular  athletic  field  is  780  feet. 
Its  length  is  5  yards  less  than  twice  its  breadth.  Find  the 
dimensions. 

32.  In  a  rectangle  24  feet  broad  and  30  feet  long  a  grass- 
plot  is  to  be  laid  out,  surrounded  by  a  flower  bed  of  uniform 
width.  It  is  desired  that  the  perimeter  of  the  grassplot  be 
exactly  one  half  that  of  the  entire  rectangle.  How  wide  should 
the  flower  bed  be  made  ? 

33.  The  value  of  15  pieces  of  money,  consisting  of  nickels 
and  dimes,  is  90  cents.    Find  the  number  of  each. 

Solution.   By  the  conditions  of  the  problem, 
the  value  of  the  dimes  +  the  value  of  the  nickels  =  90  cents. 

Let  d  =  the  number  of  dimes. 

Then  15  —  c?  =  the  number  of  nickels. 

Now  10  d  =  the  value  of  the  dimes  in  cents, 

and  5  (15  —  d)  =  the  value  of  the  nickels  in  cents. 

Therefore       10  (^  +  5  (15  -  ^)  =  90.  (1) 

Solving  (1),  c?  =  3,  the  number  of  dimes, 

and  15  —  6?  =  12,  the  number  of  nickels. 

Check.  3  •  10  +  12  .  5  =  30  +  60  =  90. 

34.  The  value  of  35  coins,  consisting  of  dimes  and  quarters, 
is  $6.50.    Find  the  number  of  each. 

35.  The  value  of  30  coins,  consisting  of  nickels  and  dimes, 
is  $2.60.    Find  the  number  of  each. 

36.  A  collection  of  nickels,  dimes,  and  quarters  amounts  to 
$10.80.  There  are  5  more  nickels  than  dimes,  and  the  number 
of  quarters  is  double  the  number  of  nickels  and  dimes  together. 
Find  the  number  of  each. 


CHAPTER  X 

DIVISION 

47.  Division  of  monomials.  The  rule  for  division  of 
numerical  terms  was  stated  on  page  29. 

Just  as  2-7-3  is  written  |,  so  a-r-h  may  be  written  as 

a  fraction,  -  * 

0 

Similarly,  a^  -^  x^  =  —-, 

x^ 

and  2  a  -*-  3  6  =  -— - . 

00 

But  12c2-4&2  =  ±f4...if_. 

4  62         52 

By  the  definition  of  an  exponent  (p.  9), 

a^  =  a  '  a  *  a  '  a  '  a  and  (jfi=a  -  a. 

Then  aB^a2=^-L^JLil±Lf=«3,  or««-l 

^  •  i 

Similarly,    2^^  2^=  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  =  2«,  or  2«-3, 

and  ao^ -r-  a;^  =  — - — 4— ^  =  ^^9  or  a2J^~ 2. 

In  like  manner    6  hi/' -^  2  1/=  3  5^^^  or  3  5  .  i/^"^. 

These  examples  illustrate  the 

Principle.  The  exponent  of  any  letter  in  the  quotient  is 
equal  to  its  exponent  in  the  dividend  minus  its  exponent  in 
the  divisor. 

RE  87 


88  FIRST  COURSE  IN  ALGEBRA 

The  foregoing  principle  expressed  in  general  terms  is 

What  this  equation  means  when  b  ~  a  and  when  b  is  greater  than 
a  will  be  explained  later. 

From  what  urecedes  we  see  that  ax^-^  x^  =  ' — ^^  /■  =  a. 

Hence  a  letter  which  has  the  same  exponent  in  divisor 
and  dividend  should  not  appear  in  the  quotient. 

The  law  of  signs  in  division  may  be  indicated  as  follows : 


-{-  ab 
+  ab 


(+  a)  -  -}-  b. 
Q-a)  =  -b. 


■  ab  -r-  (+  a)  =  —  b. 
ab  -T- (^—  a)  = -{- b. 


Now  —12  a  -^  Qb  = Here  the  quotient  is  a  fraction,  and 

b 

the  minus  sign  indicates  that  the  fraction  is  negative. 
Similarly,  d  x  -^  (—  o  y)  — > 

and  -  24  a^y  -^  (- Q  z^)  = -\-  ^. 

For  the  division  of  monomials  we  have  the 

Rule,  Divide  the  numerical  coefficient  of  the  dividend  by 
the  numerical  coefficient  of  the  divisor^  keeping  in  mind  the 
rule  of  signs  for  division. 

Write  after  this  quotient  all  the  letters  of  the  dividend  except 
those  having  the  same  exponent  in  divisor  and  dividend^  giving 
to  each  letter  an  exponent  equal  to  its  exponent  in  the  dividend 
minus  its  exponent  in  the  divisor. 

If  there  are  any  letters  in  the  divisor  unlike  those  in 
the  dividend^  write  them  under  the  preceding  result  as  a 
denominator. 


DIVISION  89 


ORAL   EXERCISES 
Perform  the  indicated  division : 

1.  -20-2. 

2.  12-5- (-3). 

3.  _36-f-(-4). 

4.  ^a^-^a\ 

5.  ce*  ^  a^ 

6.  -a^'^-^a^. 

7.  x^---(~x). 

8.  -x'-^(-x^). 

9.  ax^  -^(—  ax). 

10.  -  «^V  ^  «^2^. 

11.  —  a^xc  -7-  ca?. 

12.  a^xV  -J-  tt^a?. 

13.  -  85i^-^2a2. 

14.  9x^-^(-3x). 

^^     15  .T  V 

29.  ^^^-^-  31. 

^^    42  xV 

30.  — ;r-fi;'  32. 


15. 

_18x^-(_6x^). 

16. 

—  35  ax^  -=-  5  ax^o 

17. 

12  ax^  -  (-  3  te*^). 

18. 

14a8/;-.(-2aZ^). 

19. 

-28av^-r-(-7c?y^). 

20. 

-28c^W-^(-4aV). 

21. 

70a^yH-(-14ccy). 

22. 

16  6^c'a?.-v-2a:rc2. 

23. 

48a.x^^(-16Z»a:^). 

24. 

-36xy-r-(-6xy). 

25. 

63c^d'-^(-9cd^). 

26. 

64:a'b'  ~(-16ab'). 

.  27. 

-  28  a%^^  -i-(-7  a%y 

28. 

-  36  a^xb'  --  9  a«xZ»'. 

-17  a%' 

,,     -11  aWc^' 

34:  a'b' 

39  ^V' 

,    49«^W^ 
.^4.   ' 

-6xy  -13:rV'  Ta^xW 

48.  Division  of  a  polynomial  by  a  monomial.  In  multi- 
plying a  polynomial  by  a  monomial  (p.  73)  we  multiply 
each  term  of  the  polynomial  by  the  monomial.  In  division 
of  a  polynomial  by  a  monomial  we  reverse  the  steps  of 
multiplication  and  divide  each  term  of  the  polynomial  by 
the  monomial. 

rT^^                       ^           7   N              ax      hx  ,   J 

Thus  (ax  -{-hx)-^x  = [-  —  =  a  +  b, 

XX 

4gam,     -—^ =  —2x^-^4:X—b, 

^  -3x^ 


90  FIEST  COURSE  IN  ALGEBRA 

Therefore,  for  the  division  of  a  polynomial  by  a  monomial 
we  have  the 

Rule.  Divide  each  term  of  the  polynomial  hy  the  monomial 
and  write  the  partial  quotients  in  succession. 

ORAL  EXERCISES 

Perform  the  indicated  division  : 

a^  -  a^  9a-lSa^  9ax^-12aV 

a^c  —  a^  8  0?  — 12  0?^  ac  -\-  ah 

a^  '        —  4tx  'a 

3. 6. 9.  ^ ^. 

2x  2  ax  —  oxy 

10.   Tin 13. -^ 

4  hx^  a^G 

14  xY  -  28  xY  ax^  -  hx^  +  cx^ 

^^'  IxY  '  -x" 

4.xHj-^xY^12xY  15a'h^-\-9a%''-^0a%'' 

^^-  4xV  *  -^aV 

16a^5^-24  6i^Z>^-48a^^^ 
1^-  -%a%^ 

85  xyz  -  51  xh/z"  + 102  x^yz^  - 170  xYz 
+ 17  xyz 
4(a;  +  3)+^(a^  +  3)  Sx(3x  +  4.) -4.y(3x -{-4r), 

x  +  3  '  3^  +  4 

2(a;+l)4-3cc(a;4-l)       ^^    (^ -i>)g -(g  -  Z>)a, 

ly.  — ;  •       .00.  , 

X  +1  (X  —  6 

(«  +  x)-2fa  +  c«y  21(a;-yy-35(a;-y)° 

'^-  a  +  x  '  ''*•  -7(a;-2/)^ 

(a  +  &)«-3(«  +  ^'/  „.     -5(«c''-2a;)Ha;(a.c''-2t^) 
2^-              (a +  6)^            ■  5(ac^-2c^) 


DIVISION  91 

49.  Division  of  one  polynomial  by  another.  The  process 
of  dividing  one  polynomial  by  another  is  illustrated  in 
the  following: 

EXAMPLES 
1.  Divide  x^  —  5  x  -^  6  hj  x  —  3. 


Solution,    a:^  —  5  a:  +  I 


X  —  8  =  Divisor  Check,    a:   ~  8 


X  —  2  =  Quotient  x  —  2 


■2x  +  Q  x^-3x 

•2a:  .+  6  -2x  +  6 


a;2  —  5  a;  +  I 
2.  Divide  16x -\-12x^ -15  -  22x'' hj  2x  -  3. 


Solution.    12  a:3  -  22  a:2  +  16  a;  -  15 
12  x^  -  18  x^ 

-  4  a;2  +  16  a: 

-  4a:2+    Qx 


Check. 


2  a;  —  8  =  Divisor 


6a:2  —  2a;  +  5  =  Quotient 


+  10  a; -15 

+  10  a: -15 

6a;2-2a:  +  5 

2a:   -3 

12a:S-    4:x''  +  10x 

-18a:2+    6  a:- 

-15 

12  a:3  -  22  a:2  +  16  a:  -  15 

The  method  of  dividing  one  polynomial  by  another  is 
expressed  in  the 

Rule.  Arrange  the  dividend  and  the  divisor  according  to  tjie 
descending  (or  ascending^  powers  of  some  common  letter^  called 
the  letter  of  arrangement. 

Divide  the  first  term  of  the  dividend  hy  the  first  term  of  the 
divisor  and  write  the  result  as  the  first  term  of  the  quotient. 

Multiply  the  entire  divisor  hy  the  first  term  of  the  quotient^ 
write  the  result  under  the  dividend^  and  subtract^  being  careful 
to  write  the  terms  of  the  remainder  in  the  same  order  as  those 
of  the  divisor. 


92  FIEST  COUESE  IN  ALGEBRA 

To  find  the  second  term  of  the  quotient,  divide  the  first 
term  of  this  remainder  hy  the  first  term  of  the  divisor,  and 
proceed  as  before  until  there  is  no  remainder,  or  until  the 
remainder  is  of  lower  degree  in  the  letter  of  arrangement 
than  the  divisor. 

Check  for  division.  The  product  of  the  divisor  and  the 
quotient  should  equal  the  dividend. 


EXERCISES 

Divide  the  following  and  check  Exercises  1,  2,  3,  12,  17, 
and  18 : 

1.  ^2  4.io^_|_24byic  +  4.  9.  A:x'-%ax-\-^a''hj2x-^a, 

2.  x'-2x-16\)j  x  +  ^.  10.  x^-%x'-\-Qx-\-12hjx-2, 

3.  x^-\-x-^hj  x-2.  11.  Sx^-12x^-{-Qx-lhj2x-'l. 

4.  2>x'+bx-\-2hjx  +  l.  12.  x^-llx-Q>  by  x-\-?,. 

5.  2a2-3a-2by2a  +  l.  13.  x"" -14.x  -  d>hj  x  -  4., 

6.  5ic2-22cc  +  8byct--4.  14.  x^-2x'' -^x  +  Qhj  x-^, 

7.  Q>x'+19x-lhj?,x-l,  15.  x^-bx-\-2hj  x''  +  2x-l. 

8.  Sx'^—ax  —  2a^hjx-a.  16.  x^  — llx  + 6  by  x'-f-Sa^ -2. 

17.  x''-llx'^2x  +  12hj  x''-\-2x-4.. 

18.  a;^  +  8  by  a^  +  2.    '  20.  21  x^  -\- Sa^  hj  ^x +  2  a. 

19.  ^x^'  +  lhj  2x  +  l.  21.  ax-\-?>a-hx-3hhjx  +  :d. 

22.  S  ax  —  ay  —  6  bx  -\-  2  hg  hj  2  h  ~  a. 

23.  2Sax  -{-9ng  —  21  ag  —  12  7i.x  by  3  y  —  4  x. 

If  there  is  no  remainder,  we  have  seen  that  the  process 
of  division  may  be  expressed  as  follows : 

Dividend      ^     ,.     , 

-— — -^ =  Quotient. 

Divisor 


DIVISION  93 

If  there  is  a  remainder,  the  following  relation  holds: 

Dividend      -r»    ,.  i         ,.     ^  .  Remainder 

_  .   . =  rartiai  quotient  H ^.   . 

Divisor  Divisor 

This  last  corresponds  to  what  is  done  in  arithmetic  in 
dividing  1 7  by  5,  which  is  written  -Lt  =  3|.  This  means 
that  Jg-t.  =  3  + 1,  the  plus  sign  being  understood. 

General  check  for  division,  (a)  When  the  division  is  exact. 
Multiply  the  divisor  by  the  quotient.  The  product  should 
be  the  dividend. 

(6)  When  there  is  a  remainder.  Multiply  the  divisor  by 
the  partial  quotient  and  add  the  remainder  to  the  product 
obtained.    The  result  should  be  the  dividend. 

Note.  We  saw  on  page  1  that  it  is  customary  to  represent  the 
product,  of  two  letters  by  placing  one  after  the  other  with  no  sign 
between  them.  Thus  ah  means  a  times  h.  But  addition,  not  multi- 
plication, is  implied  by  placing  the  fraction  f  after  the  number  3. 
This  practice  comes  down  to  us  from  the  Arabs,  who  denoted  all 
additions  by  placing  the  number  symbols  in  succession  without  any 
sign  of  operation.    The  later  Greeks  also  had  the  same  notation. 


EXERCISES 
Divide  the  following  and  check  Exercises  1,  7, 11, 14,  and  22  : 

1.  6aj2-13a;  +  6by  2cc-3. 

2.  25cc^  +  30;r2-7by  5a;2-f  7. 

3.  6:z;2 +110^-35  by  2cc  +  T. 

4.  19  a  +  12  a^  -  21  by  4  «^  -  3. 
Hint.    Rearrange  the  terms  in  the  dividend. 

5.  -8  +  ic«  +  4i»-2cc2by  ^--2. 

6.  w'-Za%  +  ^ah''-h^hj  a-h. 

7.  x^-15x'-\-Q>^x  +  ^Zhjx-l, 

8.  ^x''-\-^x-2bx^-lhj  5x^-1, 

9.  2a:«-14a;2_^14cc  +  12by  2cc-4. 


94  FIRST  COURSE  IN  ALGEBRA 

10.  6  aj8  -  18  a;  +  12  by  3  cc  -  3. 

11.  4:X^  +  16x-{-12hjx^-hx-3. 

12.  6x''-2x^-}-10x-+'lShj  Sx''-4:X-\-9, 

13.  Sa^-h2Sa^-\-29a-lA0hj  Sa-  5. 

14.  -S7x-6x^-24t-{-23x''hj2x-S. 

15.  2x^  -{-lOxy  -  Ax  -  20yhy  2x  -{-lOy, 

16.  4:X^-]-Scx  +  Sax-{-6achj  4:X  +  Sc. 

17.  5Sa-{-S-53a^^l2a^hj  Aa'^Ta-l, 

18.  -  15  a^  +  56  ^2  -  99  6t  +  70  by  3  a^  -  7  a  +  10. 

19.  Sx^  +  llx^  -  Sx''  +  17  X  -  4thj  X  -  Sx''  -  4:. 

20.  6 a^i  + 15 cc?/  —  10 riy  —  9 ace  by  3x  —  2n. 
21..  23a2  +  a*-55a  +  lla3-140by  ^2-5. 

22.  25ir^-10aj2  +  36x  +  72by  5x-6. 

23.  4a3  +  l  +  a^4-4a  +  6a2by  l  +  a'  +  2a. 

24.  40  a;  -  31  cc^  +  21  +  a?*  -  4a;^  by  0^2  _  3  _  7 ^^ 

25.  a*-8a^  +  24a2-32a  +  16by  ^2-4^  +  4. 

26.  224x  -  42cc2  +  lo^;^  _  27 a;^  -  419  by  9  +  2a;2  -  5ic. 

27.  aa?  —  6  cy  —  2  ^cc  4-  4  &y  +  3  ca?  —  2  ^3/  by  2  y  —  X. 

28.  cc^  +  aV  +  a^  by  x^  —  ax  -^  a\ 

29.  a*  +  4  Z>*  +  3  a%''  by  2  ^>2  +  «^2  __  ^^^ 

30.  x^  -  4.xy  -\-St/hjx^-h2y^-{-2xy, 

31.  9a^  +  4:9b^-{-29a'b^hj  7 b^  +  3 a^ -{- 4: ab, 

32.  ic^  —  2/^  by  a;  —  y.  38.  a;^  +  ^/^  by  cc  +  2^. 

33.  a^  -  125Z>2  by  a  -  5^».  39.  a;^  +  /  by  a;  -  y. 

34.  a;*  —  16  by  cc  +  2.  40.  a;^  —  ;?/^  by  aj  +  y. 

35.  aj^-1  by  x  -1.  41.  i«^  -  y^  by  aj  —  y. 

36.  a«  +  343^»«  by  ^2  +  75.  42.  27 x« +  871^ by 3x^  +  2  71*. 

37.  8a«- 64aj^by  2a-4a;8.  43.  y' -  5^/' -  3000  by  y  -  5, 


CHAPTER  XI 

EQUATIONS  AND  PROBLEMS 

50.  Equations  involving  literal  coefficients.  The  most 
general  form  of  an  equation  in  one  unknown  is  that  in 
which  the  unknown  occurs  with  literal  coefficients.  The 
simplest  general  form  is  ax  =  b^iD.  which  x  is  the  unknown. 
The  solution  of  such  equations  involves  no  new  principle. 
It  is  merely  necessary  to  perform  the  usual  operations  with 
letters  instead  of  with  ordinary  numbers.  This  should 
cause  no  difficulty,  since  in  algebra  the  letters  are  really 
nothing  but  symbols  for  numbers,  and  should  be  used  in 
all  operations  as  freely  as  if  they  were  integers. 

In  solving  equations  whose  coefficients  involve  letters 
the  answers  obtained  will  usually  involve  these  same  let- 
ters. Only  in  exceptional  cases  may  one  expect  to  obtain 
the  root  of  such  an  equation  in  purely  numerical  form. 

In  the  following  exercises  the  unknown  is  represented 
by  a  letter  near  the  end  of  the  alphabet,  as  x  or  y,  while 
the  letters  in  the  coefficients  and  known  terms  are  taken 
from  the  beginning  of  the  alphabet. 

ORAL  EXERCISES 
Solve  for  the  unknown  : 


1. 

X  —  a  =  0. 

6. 

X  —  a  =  b. 

11. 

hx  =  ah. 

2. 

x-2c  =  3c. 

7. 

Sx  =  6a. 

12. 

ax  =  a^  +  a. 

3. 

z+5b=7L 

8. 

5y  =  10c. 

13. 

2y  =  2a  +  4:h 

4. 

X  —  a  =  a. 

9. 

7z  =  21b. 

14. 

Say  =  6  ah. 

5. 

X'hSa  =  a. 

JO, 

ax  =  a. 
95 

15. 

3a;  =  6a  +  35. 

96  FIRST  COURSE  IN  ALGEBEA 

16.  2x  —  6a=:  4:b.  19.   (a  +  b)x  =  a -\- b. 

17.  aij-~ab  =  2a.  20.   (a  +  b)y=(a-{-  b)\ 

18.  2ay  +  4.ab  =  10ab.  21.   {a -b)x=  (a -b){a  +  b). 

22.  (a  -f-  ^)  (a  —  c)a^  =  («^  +  b) (a  —  c). 

23.  (a  —  b)(a-{-b)x  =  2ab(a  +  b)(a-b). 

EXERCISES 

Solve  for  the  unknown,  and  check : 
1.  Sx  —  a  =  X  -\-  5  a. 


Solution. 

3x— a  =  x  +  6a. 

Transposing, 

3x  —  x  =  5a-{-a. 

Combining, 

2x  =  Qa. 

Dividing  by  .2, 

x  =  3a. 

Check. 

3'3a-a  =  3a  +  5a, 

or 

8  a  =  8  a. 

2.  bx  +  b  =  4:b.  11.  be  —  ex  =  4: be, 

3.  x  +  a==a-\-b.  12.  5(b  +  x)  =  10b, 

4,,  y-b  =  a-b.  13.  6(e-x)-i-lSe  =  0, 

5.  Sax  -\-  4:a  =  7  a,  14.  bx  —  (b  -\-  c)  =  5b  —  c. 

e.  ez  +  c^  =  6e'^,  15,  x  +  2  (b  -  c)  =  4:C  ■}- 2b. 

7.  ax  -\-  ab  =  ae.  16.  Sax  —  ab  =^2 ax  —  ac, 

8.  4.bx-b''=5b\  17.  4.cy-~Sac  =  5ae  +  2ey. 

9.  aV4-3caj  =  7aV.  18.  3c7/4-2^»c=  6  Jc -f  2  c?/- 3ac. 
10.  ha'x-\-Q>a'  =  a\  19.  4  ^>y  -  7  a^^  =  6  a^^  _^  3  by, 

20.  ax  —  4  a  =  (x^  +  4  —  2  a;. 

Solution.  aa:  —  4  a  =  a^  4-  4  —  2  a:. 

Transposing,  ax  ■\-  2  x  =  a^  ■\-  4:  a  -\-  ^. 

Writing  the  coefficients  of  x  as  a  binomial, 

(a  +  2)a:  =  a2  +  4a  +  4. 


EQUATIONS  AND  PROBLEMS  97 

Dividing  both  members  by  the  coefficient  of  ar, 

a2  ^  4  a  +  4 

X  — 

a  +  2 

Performing  the  division,        a:  =  a  +  2. 

Check.  Substituting  a  +  2  for  x  in  the  original  equation, 

a  (a  +  2)  -  4  a  =  a2  +  4  _  2  (a  +  2). 
Expanding,  a^  +  2  a  —  4  a  =  a^  +  4  _  2  a  —  4, 

or  a^  —  2  a  =  a^  —  2  a. 

21.  ax  -{■  hx  =  ac  ■\'  he.  23.  a^x  -f  1  —  et^  —  cc  =  0. 

22.  5  ax  -{-  4:  ex  ==  5  ab  -i-  4:  ch.        24.  ax  -^2  ah  =  2a^  -\-  hx, 

25.  ax  —  a^  —  4:  =  3a  —  X. 

26.  ace  —  ac  +  ^c  =  2  ac  —  5  5c  +  2  ^cc. 

27.  4^>V4-(a  +  ^a^)c=(a-Z>a;)c. 

Hint.    Perform  the  indicated  multiplications  first. 

28.  (x  +  a)(x-i-b)  =  x^-{-2a^  +  Sah. 

29.  15(x  -  a)-  6(x  -i-  a)=  3(5  a  -  Sx), 

51.  Uniform  motion.  If  a  man  walks  for  8  hours  at  the 
rate  of  3  miles  per  hour,  he  will  walk  in  all  8  x  3,  or 
24,  miles.  If  a  train  runs  for  12  hours  at  the  rate  of  45 
miles  per  hour,  the  total  distance  traversed  is  12  x  45,  or 
540,  miles.  These  examples  illustrate  uniform  motion.  In  all 
problems  of  uniform  motion,  the  elements  involved  are 

(a)  Time,  measured  in  seconds,  minutes,  hours,  etc. 

(5)  Rate  of  motion  (velocity),  or  the  distance  traveled 
in  a  unit  of  time  (one  second,  one  hour,  one  day,  as  the 
case  may  be). 

(c)  Distance  (total),  measured  in  feet,  inches,  miles, 
meters,  etc. 

If  a  body  moves  uniformly  for  a  time  t,  at  a  rate  r, 
and  covers  m  all  a  distance  d,  then  the  numbers  repre- . 
sented  by  c?,  r,  and  t  are  connected  by  the  equation 

d=rxt.  (1) 


98  FIEST  COURSE  IN  ALGEBEA 

This  relation  or  formula  gives  an  insight  mto  the  power 
of  the  algebraic  method.  By  means  of  arithmetical  num- 
bers alone  we  can  express  the  relation  which  holds  between 
the  time,  the  rate,  and  the  distance  only  for  a  particular 
case.  By  means  of  the  literal  equation  (1)  we  express  the 
relation  which  is  true  not  merely  for  one  case,  but  for 
countless  cases.  In  fact,  it  holds  whenever  we  are  dealing 
with  uniform  motion,  whatever  numerical  values  d^  r, 
and  t  may  have. 

The  use  of  letters  enables  us  in  this  way  to  express  a 
law  in  general  terms,  and  hence  to  include  in  one  short 
expression  a  more  perfect  idea  of  the  relation  implied  by 
uniform  motion  than  could  be  given  by  many  equations 
involving  only  arithmetic  numbers. 

ORAL  EXERCISES 

1.  Sound  travels  in  air  about  1100  feet  a  second.  A  soldier 
observes  the  shower  of  earth  thrown  up  by  an  exploding  shell, 
and  10  seconds  later  hears  the  sound  of  the  explosion.  How 
far  was  he  from  the  place  where  the  shell  struck  ? 

2.  A  man  observing  a  woodman  fell  a  tree  hears  the.  sound 
of  the  ax  three  fifths  of  a  second  after  the  blow.  How  far 
apart  are  the  two  men  ? 

3.  How  far  does  an  automobile  travel  if  it  runs 
(a)  18  miles  per  hour  for  5  hours  ? 

(h)  16  miles  per  hour  for  h  hours  ? 

(^c)  m  miles  per  hour  for  h  hours  ? 

(d)  20  miles  per  hour  for  t  -{-  4^  hours  ? 

(e)  2  a?  —  4  miles  per  hour  for  t  hours  ? 

4.  What  is  the  rate  of  an  automobile  if  it  runs  uniformly 
(a)  200  miles  in  8  hours  ?      (c)  m  miles  in  h  hours  ? 

.   (h)  m  miles  in  8  hours  ?         (d)  200  miles  in  x  +  3  hours  ? 


EQUATIONS  AND  PEOBLEMS  99 

5.  An  automobile  travels  a  distance  c?  at  a  rate  of  20  miles 
an  hour.  Another  car  runs  50  miles  farther  at  25  miles  an 
hour.  Eepresent  the  number  of  hours  required  by  each  car  for 
its  trip.  If  the  first  car  required  the  same  time  as  the  other, 
what  equation  involving  d  can  be  formed  ? 

6.  In  a  naval  engagement  the  opposing  battle  cruisers  were 
fighting  at  a  distance  of  ten  miles.  If  the  average  velocity  of 
the  shells  was  2640  feet  per  second,  how  many  seconds  did 
they  remain  in  the  air  ? 

7.  An  automobile  runs  20  miles  per  hour  for  t  hours,  and 
a  second  one  runs  14  miles  per  hour  for  ^  +  2  hours.  Eepre- 
sent the  distance  each  travels.  Form  an  equation  which  states 
that  the  distances  traveled  by  each  are  the  same. 

8.  James  travels  t  miles  per  hour  for  8  hours,  and  John 
travels  ^  —  3  hours  at  the  rate  of  12  miles  per  hour.  Eepresent 
the  distance  each  travels.  What  equation  in  t  may  be  formed 
if  both  travel  the  same  distance  ? 

9.  James  travels  from  A  to  B  in  ^  hours  at  12  miles  an 
hour.  John  leaves  A  just  2  hours  after  James,  and  traveling 
16  miles  per  hour  reaches  B  at  the  same  time  as  James.  Eep- 
resent the  distance  James  travels,  the  distance  John  travels, 
and  state  the  equation  in  t  which  may  be  formed. 

10.  Two  automobiles  leave  two  towns  225  miles  apart  at 
the  same  time  and  travel  toward  each  other.  One  travels 
m  miles  per  hour,  while  the  other  travels  5  miles  per  hour  less. 
They  meet  in  5  hours.  Eepresent  the  time,  the  rate,  and  the 
distance  for  each.  What  will  represent  the  sum  of  the  dis- 
tances traveled  ?    What  equation  in  m  may  be  formed  ? 

11.  Two  cars  run  200  miles,  one  in  t  hours,  the  other  in 
1  hour  less  time.  Eepresent  the  rate  of  each  car  in  miles  per 
hour.  If  the  rate  of  one  is  4  miles  per  hour  more  than  the 
other,  what  equation  in  t  can  be  formed  ?  If  the  rate  of  one  is 
twice  the  rate  of  the  other  ? 


100 


FIRST  COURSE  IN  ALGEBRA 


EXAMPLES  IN  UNIFORM  MOTION 

1.  A  pedestrian  traveling  4  miles  per  hour  is  overtaken  14 
hours  after  leaving  a  certain  point  by  a  horseman  who  left  the 
same  starting  point  8  hours  after  the  pedestrian.  Find  the 
rate  of  the  horseman. 

Solution.  This  is  a  problem  in  uniform  motion,  involving  the 
distance,  the  rate,  and  the  time  of  a  pedestrian  and  of  a  horseman 
respectively.  By  a  careful  reading  of  the  problem  one  discovers  that 
the  time  for  each  was  a  different  number  of  hours,  that  each  went 
at  a  different  rate,  but  that  each  traveled  tlie  same  distance.  Hence 
the  equation  will  be  formed  by  expressing  d  in  terms  of  r  and  t  for 
both  the  pedestrian  and  the  horseman  and  then  equating  the  two 
expressions  for  d. 

By  the  conditions : 


t,  or  time  in  hours 

r,  or  rate  in  miles 
per  bojir 

Distance  in  miles, 

Pedestrian 

14 

4 

50  =  4  X  14 

Horseman 

6 

X 

0  .  X 

Hence 
and 

Check. 


iS  X  —  50, 
X  =  9^. 


:50. 


BiOGiiAPHicAL  Note.  Sir  Isaac  Newton.  Sir  Isaac  Newton  (1642- 
1727)  was  probably  the  keenest  mathematical  thinker  who  ever  lived. 
He  was  the  son  of  a  farmer  of  slender  means,  and  as  a  boy  was  rather 
lazy.  It  is  said,  however,  that  his  complete  victory  over  a  larger  boy  in  a 
fight  at  school  led  him  to  feel  that  perhaps  he  could  be  equally  success- 
ful in  his  studies  if  he  really  tried.  His  ambition  and  interest  being  once 
roused,  he  never  ceased  to  apply  himself  during  the  rest  of  his  long  life. 

His  most  important  scientific  achievement  was  the  discovery  and  verifi- 
cation of  the  laws  of  motion.  In  his  great  work  called  the  ''Principia'' 
he  showed  by  mathematical  reasoning  that  all  bodies,  great  and  small,  — 
the  planet  revolving  around  the  sun,  as  well  as  the  apple  falling  from  the 
tree, — follow  the  same  laws.  His  greatest  discovery  in  pure  mathematics 
was  that  of  a  method  called  the  calculus,  which  is  the  basis  of  most  of  the 
advances  in  mathematics  and  m  theoretical  physics  made  since  his  time. 


SIR  ISAAC  NEWTON 


EQUATIONS  AND  PROBLEMS 


101 


But  important  as  was  Newton's  mathematical  work,  his  most  signifi- 
cant contribution  to  mankind  was  an  idea,  —  the  idea  that  the  world  in 
which  we  live  is  not  independent  of  the  rest  of  the  universe,  but  that 
every  smallest  particle  of  matter  is  connected  with  the  most  remote  planet 
and  star ;  that  we  cannot  think  of  the  earth  as  the  center  of  all  things,  but 
that  we  merely  occupy  our  place  in  a  system  governed  by  universal  law. 

2.  Two  men,  A  and  B,  start  from  the  same  place  at  the  same 
time  and  travel  in  opposite  directions.  B  goes  twice  as  fast  as  A. 
In  9  hours  they  are  54  miles  apart.    Find  the  rate  of  each. 

One  can  conveniently  represent  the  conditions  of  this  problem 
in  the  form  of  a  table,  as  follows : 


U  or  time  in  hours 

r,  or  rate  in  miles 
per  hour 

Distance  in  miles, 

A 

9 

r 

9r 

B 

9 

2r 

18  r 

Since  the  men  are  54  miles  apart  at  the  end  of  the  given  time, 
the  sum  of  the  distances  traveled  by  A  and  B  is  54. 

Hence  9  r  +  18  r  =  54, 

or  r  =  2,  and  2  r  =  4. 

Check.       4  =  2.2;9x2  +  9x4  =  54. 

It  should  be  particularly  noted  in  choosing  letters  for 
the  unknowns  that  it  is  not  enough  to  say,  for  instance, 
let  X  equal  the  distance,  or  let  t  equal  the  time.  This 
means  nothing  unless  the  unit  of  distance  and  the  unit 
of  time  are  also  stated.  The  unkijown  distance  is  either 
a  number  of  feet,  or  miles,  or  some  other  unit  of  length, 
and  the  unknown  time  is  a  number  of  seconds,  or  hours,  or 
some  other  specific  unit  of  time.  A  similar  remark  is  perti- 
nent each  time  a  letter  is  taken  to  represent  the  measure 
of  any  quantity. 

The  student  should  make  a  table  for  each  of  the  follow- 
ing problems,  similar  to  those  given  in  the  examples. 


102  FIEST  COUESE  IK  ALGEBRA 

PROBLEMS 

In  Problems  1-7,  A  and  B  start  from  the  same  place  at 
the  same  time  and  travel  in  opposite  directions. 

1.  A  goes  6  miles  per  hour  and  B  goes  9  miles  per  hour. 
In  how  many  hours  will  tliey  be  90  miles  apart  ? 

2.  A  travels  three  times  as  fast  as  B.  In  G  hours  they  are 
120  miles  apart.    Find  the  rate  of  eac^li. 

3.  A  travels  4  miles  more  per  hour  than  B.  After  8  hours 
they  are  144  miles  apart.    Find  the  rate  of  each. 

4.  A  goes  3  miles  less  per  hour  than  B.  After  9  hours  the 
distance  between  them  is  189  miles.    Find  the  rate  of  each. 

5.  B  goes  4  miles  less  per  hour  than  A  and  travels  two 
thirds  as  fast  as  A.  Find  the  rate  of  each.  After  how  many 
hours  will  the  distance  between  them  be  180  miles  ? 

6.  A  travels  2  hours  and  stops.  B  travels  5  hours  at  a  rate 
double  A's  rate.  Then  they  are  144  miles  apart.  Find  tlieir 
rates  and  the  distance  each  has  traveled. 

7.  Both  A  and  B  travel  the  same  distance,  B  in  9  hours, 
A  in  G.  B's  rate  is  4  miles  per  hour  less  than  A's.  Find  tlie 
rate  of  each,  and  the  distance  each  traveled. 

In  Problems  8-13,  A  and  B  start  at  the  same  time  from 
two  points  192  miles  apart  and  travel  toward  each  other  until 
they  meet.    Find  the  rate  of  each: 

8.  If  they  travel  at  the  same  rate  and  meet  in  8  hours. 

9.  If  A  travels  2  miles  less  per  hour  than  B  and  they  meet 
in  12  hours. 

10.  If  B  travels  three  times  as  fast  as  A  and  they  meet  in 
12  hours. 

11.  If  they  meet  in  9  hours  and  B  travels  42  miles  more 
than  A. 


EQUATIONS  AND  PROBLEMS  103 

12.  If  they  meet  in  6  hours  and  B  goes  4  miles  more  per 
hour  than  A. 

13.  If  they  meet  in  12  hours  and  A  travels  6  miles  more  per 
hour  than  B. 

In  Problems  14-16,  A  and  B  start  at  the  sam«  time  from 
two  points  144  miles  apart  and  travel  toward  each  other  until 
they  meet.  Find  the  number  of  hours  from  the  start  until  the 
time  of  meeting : 

14.  If  B  goes  4  miles  more  per  hour  than  A  and  travels 
twice  as  far  as  A. 

15.  If  A  travels  6  miles  per  hour  and  B  travels  9  miles 
per  hour,  but  B  is  delayed  4  hours  on  the  way. 

16.  If  A  is  delayed  3  hours  and  B  is  delayed  5  hours,  and 
their  rates  are  16  miles  and  8  miles  per  hour  respectively. 

17.  The  distance  from  Kansas  City  to  St.  Louis  is  285  miles. 
A  passenger  train  running  45  miles  per  hour  leaves  Kansas 
City  for  St.  Louis  at  the  same  time  a  freight  train  running 
12  miles  per  hour  leaves  St.  Louis  for  Kansas  City.  In  how 
many  hours  will  they  meet  ? 

18.  A  starts  from  a  certain  place  and  travels  4  miles  per 
hour.  Six  hours  later  B  starts  from  the  same  place  and 
travels  in  the  same  direction  at  the  rate  of  6  miles  per  hour. 
How  many  hours  does  B  travel  before  overtaking  A  ? 

19.  Two  bicyclists  108  miles  apart  start  at  the  same  time 
and  travel  toward  each  other.  One  travels  10  miles  per  hour, 
the  other  12  miles  per  hour.  The  latter  is  delayed  2  hours  on 
the  way.  In  how  many  hours  will  they  meet,  and  how  far  has 
each  traveled  ? 

K  20.  A  passenger  train  starts  2  hours  later  than  a  freight 
train,  from  the  same  station  but  in  an  opposite  direction.  The 
rate  of  the  passenger  train  is  42  miles  per  hour  and  the  rate 


104  riEST  COUESE  IN  ALGEBEA 

of  the  freight  train  is  24  miles  per  hour.  In  how  many  hours 
after  the  passenger  train  starts  will  the  two  trains  be  246  miles 
apart  ? 

21.  A  messenger  going  at  the  rate  of  8  miles  per  hour  has 
journeyed  2  hours  when  it  is  found  necessary  to  change  the 
message.  At  what  rate  must  a  second  messenger  then  travel 
to  overtake  the  first  in  8  hours  ? 

22.  A  man  having  4  hours  at  his  disposal  wished  to  ride  as 
far  out  of  town  as  possible  on  a  trolley  car  whose  rate  is 
10  miles  per  hour,  and  to  return  on  foot  at  the  rate  of  3  miles 
per  hour.  On  the  way  back  he  can  take  a  short  cut  and  save 
one  mile.    How  long  a  time  may  he  ride  on  the  car  ? 

The  velocity  of  a  bullet  continually  decreases  from  the 
instant  it  leaves  the  gun.  This  is  due  to  the  resistance  of  the 
air.  In  the  following  problems  consider  the  velocity  of  sound 
to  be  1100  feet  per  second : 

23.  Two  and  one-half  seconds  after  a  marksman  fires  his 
rifle  he  hears  the  bullet  strike  the  target,  which  is  550  yards 
distant.    Eind  the  average  velocity  of  the  bullet. 

24.  One  and  three-quarters  seconds  after  a  marksman  fires 
his  revolver  he  hears  the  bullet  strike  the  target  50  rods  dis- 
tant.   Find  the  average  velocity  of  the  bullet. 


CHAPTER  XII 
IMPORTANT  SPECIAL  PRODUCTS 


52. 

The 

square 

of  i 

I  binomial.    The 

a  -i-b 
a  -\-b 
a^-\-     ab 
+     ab-hU^ 

multiplication 

a^-h^ab  +  b^ 
gives  the  formula 

(a+by  =  d'  +  2ab  +  V'. 

This  ma}^  be  expressed  in  words  as  follows : 

7.  The  square  of  the  sum  of  two  terms  is  the  square  of  the 

first  term  plus  twice  the  product  of  the  two  terms  plus  the 

square  of  the  second  term. 

Similarly,  (a  -  5)"  =  a"  -  2  a&  +  fe^ 

which  may  be  expressed  in  words  as  follows: 

IL  The  square  of  the  difference  of  two  terms  is  the  square 
of  the  first  term  minus  twice  the  product  of  the  two  terms  plus 
the  square  of  the  second  term. 

Study  the  application  of  I  and  II  in  the  following: 

EXAMPLES 

1.  (x  +  l)2=cc2  +  2x  +  l.  3.  (3  +  a)^=9  +  6a  +  a'. 

2.  (2^_2)2=?/2-4y  H-4.  4.  (5 - m)^ = 25- 10 m-fm". 

5.  (3  a  +  Z>)2  =  9  6^2  ^  6  aZ»  +  h\ 
105 


106  FIRST  COURSE  IN  ALGEBRA 

ORAL  EXERCISES 

Expand  the  following  by  I  or  II,  page  105 : 

1.  (a  -f  ly.  7.  (6  +  x)\  13.  (a  -  c)\ 

2.  (b  +  2)2.  8.  (9  -  df.  14.  (a  +  2  b)\ 

3.  {x  -  5)'-^.  9.  (11  -  yf.  15.  (3^^  ~  xf, 

4.     (y  4    10)2.  IQ^     (^   _^  ^^^2^  Ig^     ^5  ^.  _  ^s^2^ 

5.(^-12)2.  11.   (yy -f- ^»)2.  17.   (7  Z  4- //?-)". 

6.  (2  +  a^)2.  12.  (m  -  7i)2.  18.  (12  in  -  n)\ 

State  two  equal  binomials  whose  product  is : 

19.  m^  4- 4  m  4- 4.  22.  d""- 12  d -{- ^^. 

20.  x^  +  10x-{-  25.  23.  hj"  +  14  A  4-  49. 

21.  (*2_6a-f-9.  24.  64 -16m  4- m'. 

Find  the  value  of  the  following: 

25.  (5  4-2)1 

Solution.    (5  +  2)2  II.  52  4  2  .  5  •  2  +  2'^  ::^  25  +  20  +  4  =  49. 

26.  (10  4- 1)'.  27.  (9  +  2)2. 

28.  (10  -  3)2. 

Solution.    (10  -  3)2  =  I02  -  2  •  10  •  3  +  32  =  100  -  60  +  9  =  49. 

29.  (12  -  2)2.  33.  (53)2.  33^  ^29)2. 

30.  (14-4)2.  34^   (103)2.  Hint.   (29)2  = 

31.  (21)2.  35.  (109)2.  (30-1)2  etc. 
Hint.  (21)2  ^                         3g^   /201)2,                      ^^'   (^^)'- 

(20 +  1)2  etc.  3^  '  40.  (98)2. 

32.  (32)2.  •  ^       ;  •  ^^     ^gc)g^2^ 

EXERCISES 

Expand  the  following : 

1.  (2  c  -  3  d)\       5.  (7  a;2  +  2  ijf.  9.  (12  aV  -  3  c)\ 

2.  (px-2 y)\        6.  (5 a;2  -  3  tj)\  10.  (5  d^b  -  2  a^f. 

3.  (3  a -2^/2)2.       7.  (9^2  +  2  62^^2^  11.  {- 2 ah'' +  ^ cdy. 

4.  (4  a;  +  5  /)2.      8.  (11  a%  +  2f.  12.  (-4  xif  -  3  xz^. 


IMPOPvTANT  SPECIAL  PRODUCTS  107 

Make  any  changes  which  are  necessary  in  the  following  so 
that  each  numerator  will  be  the  square  of  its  denominator : 

x^  -\-  2xy  -^  y^  m^  -|-  16  m  —  64 

13.  ;  •  19.  ^    ;  • 

X  -\-  y  8  +  111 

m''-mn^-n^\  a^  +  10  ^'  -  25 


w.  —  n 


5 


^^    a^  4-10^  +  25  ^,     9a;^-6a;  +  l 

a  -{-b  6x  -\-l 

^       0^2  4-12^  +  36  __    4a2  +  12a/>  +  9Z>" 

16. •  22.  -  —  • 

x  —  h  La  —  6b 

4  +  4.x  +  a^^  4r^^-20:rV+.25  2/^ 

^^'         2^x       '  2x'-by 

2b-10x-Yx^  Ax' -20x^y-\- 25 tf 

5  —  X  5y  —  2x^ 

53.  The  product  of  the  sum  and  the  difference  of  two  terms. 

The  multiplication  a  4-b 

a  —h 
iifi  -h  ab 

-  ah  -  Jfl 
a^  - 1^ 

gives  the  formula  {a -{■b)(a—b)  =  a^  —  V^. 

This  may  be  expressed  in  words  as  follows: 
III,   The  product  of  the  sum  and  the  difference  of  two  terms 
equals  the  difference  of  their  squares  taken  in  the  sairie  order 
as  the  diffe7'e7ice  of  the  terms. 

The  pupil   should  study  the   application  of  III  in  the 
following : 

EXAMPLES 

1.  (^  +  1)  {x  -  1)  =  0^2  -  1.       3.  {a  -^x){a-x)=  a^  -  x^. 

2.  {lS-x)(5-\-x)  =  25-x\      ^,  (2e-y)(2c^y)  =  Ac''-y\ 

5.  (c2-3)(c2  +  3)=c*-9. 


108 


FIRST  COURSE  IK  ALGEBRA 


ORAL  EXERCISES 
Perform  the  following  indicated  operations  : 


1.  (a +  2)  (a -2). 

15. 

(2a  +  3bc')(2a-Sbc'), 

2.  (m  +  7)(m-7). 

16. 

(a  +  3a%)(a-Sa%). 

3.  (A +  9)  (7^.-9). 

17. 

(1  a +  2x)(2x -la). 

4.  (6  +  0(6-0. 

18. 

(px  +  3y){3y-bx). 

5.  (1_^)(1  +  ^). 

19. 

(5m2-47i)(5m2  +  47i). 

6.  {c-\-d)(c-d). 

20. 

(2  ah -^3  c)  (2  ah -3  c). 

7.  (x-\-^y)(x-^y). 

21. 

(9  cd''  -  2)  (9  cd^  +  2). 

8.  {^x^2y){^x-2y). 

22. 

(10mn'-3p)(10m7v'+3p). 

9.  (l-\-lx){l-4tx). 

23. 

(^ia^^9b')^(2a  +  3h) 

10.  (9  a -1)  (9  6^  +  1). 

24. 

(9x^-25y^)---(3x-5y). 

11.  (2  +  3r)(2-3r). 

25. 

(4.a^P-c^)-i-(2ah-c). 

12.  (aj2  -  4)  (:«2  ^  4). 

26. 

(49-a^^)--(7+i»2^^. 

13.  (lx'-{-2y){lx'-2 

y)' 

27. 

(25x^-4:)-^(5x^-}-2). 

14.  {4.x''+^xhj){4.x^-?,xhj). 

28. 

(36a^-49)-v-(7+6a2). 

State  two  binomials  whose  product  is : 

29.  m2-9.                   33. 

81 

-f. 

37.  25x^-4:y^. 

30.  c^-ie.                   34. 

100-7^2.                38.  9a2-16^»2. 

31.  49 -ml                 35. 

a^- 

—    4:X 

39.  36c2-l. 

32.  m2-49.                 36. 

^2- 

-9y 

\                 40.  l-49aj2 

43.  (30  +  1)  (30  -  1). 

44.  (50  -  2)  (50  +  2). 


Eind  the  value  of : 

41.  (12  -  2)  (12  +  2). 

42.  (40  +  3)  (40  -  3). 

45.  22  X  18. 

Hint.    22  x  18  =  (20  +  2)  (20  -  2)  etc. 

46.  13  X  7.      48.  43  x  37.       50.  71  x  69.     52.  96  x  104. 

47.  23  X  17.    49.  64  x  56,       51.  83  x  77.     53.  91  x  109. 


IMPOETANT  SPECIAL  PEODUCTS  109 

54.  The  product  of  two  binomials  having  a  common  term. 
The  multiplication 

X  +5 


^-{-  ax 
-i-  hx-\-  ab 


x^ -{- (a -\- H)  X -\-  ah 
gives  the  formula 

(x+  a)(;c+  &)  =  jc'  +  (a+  6)jr  +  a6. 

This  may  be  expressed  in  words  as  follows : 

/F.  The  product  of  two  binomials  having  a  common  term 
equals  the  square  of  the  common  term^  plus  the  algebraic  sum 
of  the  unlike  terms  multiplied  by  the  common  term^  plus  the 
algebraic  product  of  the  unlike  terras. 

The  pupil  should  study  the  application  of   IV  in  the 


following  ; 


EXAMPLES 


1.  {x-\-V)(x-\-2)=^x'-^{l-\-2)x-\-2  =  x'-\-^x-{-2.       '  - 

2.  (c  -  2)(c  -  3)  =  e'  +  (-  2  -  3)c  J^  Q  =.  c"  -  be  +  ^. 

3.  {n  -  5)(n  -\-  2)=  rv"  +(-  5  -\-  2)n  -10  =  n"  -  3n  -10, 

ORAL  EXERCISES 

Multiply  the  following  mentally  : 

1.  (x  +  2)(x  +  1).  9.  (x  -5)(x-  2). 

2.  (c4-3)(c  +  2).  10.  (aj-7)(;r-8). 

3.  (h-\-A)(h-}-5),  11.  (c-8)(c  +  9). 

4.  (a +  5) (a  +  3).  12.  (c  +  10)  (c  -  3). 

5.  (a-}-7)(a  +  2),  13.  (c-7)(c  +  5). 

6.  (c  +  5)(c  +  4).  14.  (x  +  a)(x-i-c), 
^7,  (x-  2)  (^  -  3).  15.  (x  -{-k)(x-^  I). 

8.  {c  -  3)  (c  -  4).  16,  {x  '\-a)(x-  c). 


110  FIKST  COUKSE  IN  ALGEBRA 

EXERCISES 

Write  the  quotients  for  the  following  by  inspection  and 
check  by  multiplication : 

x-{-3       '  *  x-\-2        ' 

a"  -f-  7  <^  +  10  q    </^  -  3  6t  -  28 

'  g^-f  11(/. -f  10  ^  f/'  ~-  3  <(/>  -f  2  li' 


7;^  -\-  '6  n 

,■'- 

7  /v^  -  18  .s^ 

r  -9s 

y"  — 

rs  -  90  s' 

(*  -H 10  '  f^  -  /> 

b  -\-l  X  —  6y 

r-^  —  6  c  4-  8  ,  ^     ^>/^  —  4  vy^r?,  —  21  n^ 

5.  ^_,       •  12. 

6.  li:zliL±ii.  13. 

C  —  i 

'  a^-2  r-10  6' 

55.  The  product  of  two  binomials  of  the  form  {ax-\-h) 
(ex  -\-  d).    The  multiplications 

6  :r2  4- 15  a;  ac':^^  -f-    ^crc 

4- 14  a;  +  35       and  +   adx  +  ^^ 

6  2:2  4. 29  a;  +  35  acx^ -h  (be  +  ad)  x -{- bd 

show  that  in  the  product  of  any  two  binomials  of  the  form 

(^ax  +  6)  (^cx  +  cZ) 

(a)  the  first  term  is  the  product  of  the  first  terms  of 

tlie  binomials, 

(6)  the  second  term  is  the  sum  of  the  cross  products,  and 
(^)  the  third  term  is  the  algebraic  product  of  the  second 

terms  of  the  bmomials. 


IMPOETAISTT  SPECIAL  PRODUCTS  111 

EXAMPLES  * 

=  12:^2^23:^  +  10. 

2.  (5a-{-l)(Sa-2)  =  15a^-\-3a-10a-2 

^  =15a^-7a-2. 

3.  (4Z>-3)(Z»-h2)=4/>2-f  5/>- G. 

I I 

EXERCISES 

Write  the  prodnots  in  the  foUowing: 

1.  (2  a  +  1)X^^  +  ^)'  11-  (7  ^''  +  ^)  (4  ^^  -  ^)- 

2.  (3  a  +  2)  (a  +  5).  12.  (5  k  -  4)  (5  A  -  4). 

3.  (:c-f-4)(2.T  +  3).  13.  (5  k  -  (j)(5k  -  Q). 

4.  (2 0^  +  3)  (x  +  7).  14.  (3  -h  m)  (4  +  2  m). 

5.  (2c  +  3)(c  +  8).  15.  (7-ri)(7-3  7i). 

6.  (4(?  +  3)(4c  +  3).  16.  (2a-b)(Sa-^2b). 

7.  (5m  +  2)(5m  +  2).     .  17.  (5a  +  9c)(2a  -  9c). 

8.  (4m-l)(2m  +  3).  18.  (2x'^ -  7 y)(Sx'' -  2^} 

9.  (5n-3)(37i-5).  19.  (c2-56Z)(5c2  +  2^). 
10.  (27i  +  7)(37i-2).  20.  (Ae  +  9fP)(2c-5cP). 

ORAL  REVIEW  EXERCISES 

Perform  the  following  indicated  operations : 

1.  (m  +  2  rf,  7.  («  +  3  c)  (a  -  3  c). 

2.  (7^  -  3  ky,  8.  (5  a  +  1)  (5  a  -  1). 

3.  (2  a +13)2.  9.  (5  7i2_y)2. 

4.  (6s -5 r)\  10.  (3 rs"-  +  2 .s)2. 

5.  {a'-^r^xf.  11.  (2a5  +  3//)(2a!-3y) 

6.  (2  m  -  3  ^2)2.  12.  (4 IM  -  5  h^f. 


112  FIEST  COUESE  IN  ALGEBEA 

13.  (5  x"^  -  z)  (5  x"  +  z).  17.  (a  +  3)  (a  +  2). 

14.  (-  5  aV  -  2  c)2.  18.  (a  -  9)  (a  -  4). 

15.  (31)1  19.  (Im'n  +  2>){l7ri'n-^). 

16.  (49)2.  20.  (A:xy'^^^z'){4cxy'-5z'^), 

21.  (^  +  2/)(cc2^2/')(^-2/). 

Hint.   Multiply  the  product  of  the  first  and  third  binomials  by  the 
second  binomial. 

22.  (a-2)(a2  +  4)(a  +  2). 

23.  (m  +  n)  (m^  +  ^^)  {^^^  —  ^)  (^^^  +  n^)- 

24.  (CC2  +  1)(:^2_1)(^4_|_;1^^^(^8_^1)^ 

25.  [(a  +  Z^)  +  c][(c^  +  Z.)-c]. 

26.  [(a.  +  2/)+2][(cc  +  2/)-2]. 

27.  (:r  +  2a)(aj  +  «).  33.  (3  a^  +  4)  (3  6^2  ^  4). 

28.  (2c  +  l)(c  +  2).  34.  (o?-i-^h){2a'-h). 

29.  (^d''-\-4.)(2d^  +  ^).  35.  (5^2-2/)(5a;2_^^^ 

30.  (x-y)(x  +  ^  y).  36.  (4  c^  +  d^)  (4  c^  -  dy 

31.  (c-3^)(c-^).  37.  (4ir2-2a«)(4a^2_3^8>j^ 

32.  (Iv"  +  2k)  Qi'  -3  k).  38.  (5 :« -^/V) (2  oj +3  yV), 

39.  (^2  +  2  am  +  m^)  -^  (<x  +  m). 

40.  (;:c2-42/2)--(a;-2  2/). 

41.  (a^  _  3  a  +  2)  --  (a  -  2).      '^' 

42.  (49— 14:c+  ^^)^(7-a;). 

43.  (c2-5c  +  6)--(c-3). 

44.  (9c2_4)-i-(3c  +  2). 

45.  (d''-16d-17)-^(d  +  l).   J. 

What  is  the  quickest  way  to  multiply : 

46.  (x  +  y)(x  +  y) (x  —  y)(x  —  y)? 

47.  (x^  +  a")  (x  +  a)  (x^  +  a')  (x-a)? 

48.  (Ax''  +  a')(2x  +  a)(16x^  +  a')(2x  -  a)?  ^ 


CHAPTER  XIII 
FACTORING 

56.  Definitions.  Factoring  is  the  process  of  finding  two 
or  more  expressions  whose  product  is  equal  to  a  given 
expression. 

Many  simple  exercises  in  factoring  were  solved  in  the 
preceding  chapter  in  connection  with  the  rules  of  multi- 
plication there  given.  In  fact  the  process  of  factoring  is 
the  reverse  of  multiplication. 

The  subject  of  factoring  is  extensive.    In  this  chapter  ' 
we  shall  consider  only  the  more  common  forms  of  factor- 
able expressions,  using  only  such  factors  as  have  integers  ' 
as  coefficients.  '"^^ 

If  a  polynomial  cannot  be    expressed   as  the  product'^ 
of   expressions   other  than  itself   and  1,  it  is  said  to  be 
prime. 

57.  Square  root  of  monomials.  In  factoring  it  is  often 
necessary  to  find  the  square  root,  the  cube  root,  and  other 
roots  of  monomials. 

The  square  root  of  a  monomial  is  one  of  the  two  equal 
factors  whose  product  is  the  monomial.  ^ 

Since  +  2  •  +  2  =  4  and  —  2  •  —  2  =  4,  the  square  root  of  4  is  ±  2, 
which  means  both  plus  2  and  minus  2.  . 

Similarly,  the  square  root  of  9  is  ±3  and  the  square  root  of  a^  i^  ±  q. 

That   is.  Every  positive    number  or  algebraic   expression  '> 

has  two  square  roots  which  have  the  same  absolute  value  but 
opposite  signs,  :  / 

113 


114  FIRST  COURSE  IN  ALGEBRA 

It  is  customary  to  speak  of  the  positive  square  root  of 
a  number  as  the  principal  square  root,  and  if  no  sign  precedes 
the  radical,  the  principal  root  is  understood. 

Thus  VI  =  2,  not  -  2  ;  -  V4  =  -  2,  not  +  2. 

When  both  the  positive  and  the  negative  square  roots 
are  considered,  the  double  sign  must  precede  the  radical. 
Since        x^  -  x^  =  (—  x^)  (—  x^)  =  x^^  then  ±  Vo;^  =  ±x^. 

That  is,  The  exponent  of  any  letter  in  the  square  root  of 
a  monomial  is  one  half  the  exponent  of  that  letter  in  the 
monomial. 

Hence  for  extracting  the  square  root  of  a  monomial 
where  both  positive  and  negative  factors  are  desired  we 
have  the 

Rule.  Write  the  square  root  of  the  numerical  coefficient  pre- 
ceded hy  the  double  sign  ±  and  followed  by  all  the  letters  of 
the  monomial^  giving  to  each  letter  an  exponent  equal  to  one 
half  its  exponent  in  the  monomial. 

A  rule  much  like  the  preceding  holds  for  fourth  root, 
sixth  root,  and  other  even  roots. 

Thus  iA/sT^  -:  ±  3  c2,  and  ±\V«  =  ±  a^. 

In  the  chapters  on  Factoring  and  Fractions,  where  square 
roots  arise,  only  the  principal  square  root  will  be  considered. 

According  to  the  definition  of  square  root,  the  two 
factors  of  a  term,  either  of  which  is  its  square  root,  must 
be  equal.    Consequently  they  must  have  the  same  sign. 

58.  Cube  root  of  monomials.  The  cube  root  of  a  mono- 
mial is  one  of  the  three  equal  factors  whose  product  is 
the  monomial. 

Since  3  •  3  •  3  =  27,  -v/27  =  3. 

And  as         -3--3.-3  =  -27,     v^- 27  =  -  8. 


FACTOEIlSra  115 

That  is,  The  cube  root  of  a  monomial  has  the  same  sign  as 
the  monomial. 

Since  3^  -  x^  -  x^  =  a:^^  V^^  =  x^. 

Similarly,  as       a^x^  •  a^x^  •  a^x^  =  a^^x^,       -Va^^x^  =  a^x\ 

That  is,  The  exponent  of  any  letter  in  the  cube  root  of  a 
term  is  one  third  of  the  exponent  of  that  letter  in  the  term. 

Hence  for  extracting  the  cube  root  of  a  monomial  we 
have  the 

Rule,  Write  the  cube  root  of  the  numerical  coefficient  pre- 
ceded by  the  sign  of  the  monomial  and  followed  by  all  the 
letters  of  the  monomial^  giving  to  each  letter  an  exponent  equal 
to  one  third  of  its  exponent  in  the  monomial, 

A  rule  much  like  the  preceding  holds  for  fifth  root, 
seventh  root,  and  other  odd  roots. 


Thus  V-32  =-2,  Vx^=x\  and  Vl28~^i  =  2x\ 


ORAL  EXERCISES 

Find  the  value  of  the  following : 
1.    V3"2.  13.  V49rV^ 


2.  Vr.  14.  VToO^i^. 


3.  Ve"^  15.  V144  ay\ 

16.  Vl96A^. 

17.  VI2I  a^h^'c\ 

18.  V2257W. 


4. 

V52 .  2\ 

5. 

V9.5I 

6. 

V4x-l 

7. 

V9^. 

8. 

Vl6^. 

9. 

■\f26x\ 

10. 

V4aV. 

11. 

V36  aV. 

19.  V289aW. 

20.  V361  cc yv^. 

21.  V225m^V^^ 


22.  V22 .  3* .  51 


25. 

V9  .  25  .  81. 

26. 

■>Jm\ny. 

27. 

■\/a\by. 

28. 

-^/4.x\yy, 

29. 

V22  .  (72)2. 

30. 

V32 .  (2^)^ 

31. 

^x'  .  {y^y. 

32. 

V25  .  2« .  r\ 

33. 

2V^. 

34. 

5Va^.2i^ 

23.  VW^¥^\  35.  rVS^Ts^.y 

12.  V64mV^       24.  V52 .  2*  •  9.  36.  -^. 


116  FIEST  COUESE  IN  ALGEBEA 


37. 

■v^(-  2f, 

44. 
45. 
46. 
47. 
48. 
49. 
50. 

■y/l2bx'^. 

51. 
52. 
53. 
54. 
55. 
56. 
57. 

■y/x^Y^ 

38. 
39. 

■v^(-8).2l 
Vb^ .  3«. 
^8  .  3«. 

-v^343  .  a^^l 
-v^(-  xf  .  51 

■^2''x^. 
-v/3*5l 

40. 

-^(_  3)6 .  82 .  c\ 
-v^lOOO  .  r\ 
■^-729.9^.5^. 

-^^16. 

41. 
42. 

V-  21 .  8. 
-^27  .  x\ 

■v^-243ai^ 

43. 

■^-64.^1 

^4  .  2^aH''', 

59.  Polynomials  with  a  common  monomial  factor.  The 
type  form  is  ah-^ac-ad. 

I    Since  ab -i-ac  — ad  =  a(h -\- c  —  d^, 

we  have,  for  factoring  expressions  having  a  common  mono- 
'mial  factor,  the  following 

Rule.  Determine  by  inspection  the  monomial  factor  which 
is  the  product  of  all  numerical  and  literal  factors  common  to 
all  terms  of  the  polynomial. 

Divide  the  polynomial  by  this  monomial  factor. 

Write  the  quotient  in  a  parenthesis  preceded  by  the  mono- 
mial factor. 

EXAMPLE 

Factor  10  ab^  -  15  b\ 

Solution.    The  common  monomial  factor  of  both  terms  is  seen 
to  be  5  b^.    Dividing  the  binomial  by  5  b^,  the  quotient  is  2  a  —  3  5. 
Therefore  10  ab^  -16b^  =  6b^(2a  -  Sb). 

ORAL  EXERCISES 
Factor  the  following : 

11.  2b-h2c.  16.  20c-{-lScd. 

12.  5r-10s.  17.  25m-20m.n. 

13.  6x-4.y.  18.  27x-36xy. 

14.  lOcc  +  2?/.  19.  30r-12r5. 

15.  16a-U.  20.  S5t-{-^st. 


1. 

2  a; +  2. 

6. 

7x-U. 

2. 

Sx-j-6. 

7. 

10  a; +  20. 

3. 

2a- A. 

8. 

5  771  — 15, 

4. 

3c +  9, 

9. 

10 -2n. 

5. 

5  c -10. 

10. 

12 -6  a. 

FACTORING  117 

EXERCISES 

Write  the  prime  factors  of  the  following : 

1.  3  ax''  +  18  a, "         12.  3  ^z'  +  6  /c  -  3  y\ 

2.  8cc«+8iK.  13.  a'-^a^-a''+2a. 

3.  2cc*-6ajl  14.  5r2  +  10r  +  157^ 

4.  aW  +  a^'b^  15.  6  a^  -  12  a*  +  6  a  -  18  a«. 

5.  3^^5-27 7^.  16.  16m«-32m27i+24mV. 

6.  10;i^^-4i^l  17.  -a%-2aV-\-a%\ 

7.  18  ic^ -{- 27  4- 9  cc.  18.  Solve  for  a^,  axe  =  a  (m  +  n). 

8.  x^  —  x^  -\-  x^,  19.  Solve  for  x^  ax  =  a&  -\-  ac. 

9.  2  c^  —  18  c  +  2  ^^.,,,,...^0.  Solve  for  y,  ti?/  =  Tir  +  ns  +  n^. 

10.  4:  x^  —  S  ax -j- 20  X,       21.  Solve  ioTy,  my =mh—mk—mL 

11.  5  a^  +  10  a^  -  25  a^     22.  Solve  for  z,az  =  2a-ab  +  a\ 

60.  Polynomials  which  may  be  factored  by  grouping  terms 
and  taking  out  a  common  binomial  factor.   The  type  form  is 

ax-^ay-^bx+by. 

Plainly     ax  +  ay  -}-  hx  -{-  by  =  a(^x  -{-  ^^  -}-  h  (^x  -\-  ^). 

Dividing  both  terms  of  a(x  -\-  y^  +  b  (x  -^  y')  by  (x  +  y),' 
the  quotient  is  a-{-b. 

Therefore     ax '\-  ay  -{•  bx -\- by  =  (x -^  y)  (a  +  &), 

EXAMPLE 

Factor  2  xy  -{- ^  ab  -\-  ^  ay  -{- bx.  ! 

Solution.  2  a:y  +  3  a6  +  6  a?/  +  fto;  = 

2xy  -\-  Qay  •\-  hx  -^  ^  ah  = 

2  y (a;  +  3  a)  +  6(a:  +  3  a)  = 

(a:  +  3  a)  (2  .V  +  ft). 


118  FIRST  COUESE  IN  ALGEBBA 

The  preceding  example  illustrates  the 

Rule,  Arrange  the  terms  of  the  polynomial  to  be  factored 
in  groups  of  two  or  more  terms  each,  such  that  in  each  group 
a  monomial  factor  may  he  written  outside  a  parenthesis^ 
which  in  each  case  contains  the  same  expression. 

Rewrite,  placing  these  monomial  factors  outside  parentheses. 

Then  divide  by  the  expression  in  parenthesis  and  write  the 
divisor  as  one  factor  and  the  quotient  as  the  other. 

Polynomials  which  may  be  factored  by  grouping  terms 
according  to  the  foregoing  rule  usually  contain  either 
four,  six,  or  eight  terms. 

It  is  important  to  note  that  one  can  obtain  two  apparently  differ- 
ent sets  of  factors  for  a  given  expression.    Thus 

(m  —  r)  (n  —  2  a:)  =  (r  —  m)  (2  a;  —  w), 

for  each  pair  by  actual  multiplication  gives  mn  —  nr  —  2  mx  ■{•  2  rx. 

An  inspection  of  the  expression  shows  that  the  binomials  of  the 
first  pair  are  the  negatives  respectively  of  those  in  the  second  pair ; 
hence  either  pair  of  factors  is  correct. 

The  relation  that  the  process  of  factoring  bears  to  the 
processes  of  multiplication  and  division  of  monomials  and 
polynomials  should  be  constantly  kept  in  mind.  In  multi- 
plication we  have  two  factors  given  and  are  required  to 
find  their  product.  In  division  we  have  the  product  and 
one  factor  given  and  are  required  to  find  the  other  factor. 
In  factoring,  however,  the  problem  is  a  little  more  difficult, 
for  we  have  only  the  product  given,  and  our  experience 
is  supposed  to  enable  us  to  determine  the  factors.  For 
this  reason  a  very  careful  study  of  various  typical  forms 
of  products  is  necessary. 

There  is  no  simple  operation  the  performance  of  which 
makes  us  sure  that  we  have  found  the  prime  factors  of  a 


FACTOEmG  119 

given  expression.    Only  insight  and  experience  enable  us 
to  find  prime  factors  with  certainty. 

A  partial  check,  however,  that  may  be  applied  to  all 
the  exercises  in  factoring,  consists  in  actually  multiplying 
together  the  factors  that  have  been  found.  The  result 
should  be  the  original  expression. 

ORAL  EXERCISES 

Factor  the  following : 

1.  5{x-\-y)+c{x  +  y).  7.  ^x{2x-y)-^y{2x-y), 

2.  a{2h  +  c)-{-c(2h  -\-c).  8.  m(m—5n)—2n(m  —  5n), 

3.  m(a-h3x)--n(a  +  Sx).        9.  2  a(a  —  h)— Sb(a  —  b). 

4.  2x{x-2y)-y{x-2y).      10.  5cc(a;  -  3)+ 3(ic  -  3). 

5.  3r(5+7^)-(5+70.  11.  4r(r-7)-7(r-7). 

6.  2m(m-\-n)-^?>n{n^m).      12.  1  s{s  —  t)—2t(s  —  t), 

13.  9 m  (m  —  2  n)  -{-  4:  n(m  —  2  n). 

14.  7x(3x-4.tj)-2y(3x-4.y), 

15.  llab(a-Sh)-5c(a-Sb). 

EXERCISES 
Factor  the  following : 

1.  2a(a  —  x)-{- 3(x  —  a). 

Hint.   This  can  be  written  2  a  (a  —  x)  —  3  (a  —  a). 

2.  7m(m  —  2n)—2(2n  —  m), 

3.  3h(5  h  -  k)-  k(k  -  5  h)  -  hk(5 h  -  k), 

4.  7x(2x-5y)-2y(2x-5y)-{-(5y-2x). 

5.  (Sa  -  2x)-{-  9 a(-  2x  +  3a)-{-  x(2x  -  3a), 

6.  ah -{- bh -\- ak  -{-  bk. 

R£ 


120  FIEST  COUESE  IN  ALGEBEA 

7.  mr  -i-  nr  -\-  ms  -\-  ns.  13.  mh  —  nh  -\-  mk  —  nk, 

8.  2aG-\-hG-{-2ad-\-hd.  14.  2ax —  ^ay  ■\-hx  —  Z  hy, 

9.  3  ac  +  c^  +  9  ay  +  3  xy.        15.  ar  —  cr  -\-  as  —  cs. 

,r    10.   6  ax -{- 15  bx-\-4t  ay -\-10  by,      16.  mr  —  nr -j- ms  —  ns. 

11.  2a^  +  a2a^  +  2acc+a^l  17.  2  hx  -  2kx  +  hy  -  ky. 

12.  6A^  +  4A2A'+3M--f  2;bl     18.  3  x^  -  3  cc^  +  a^y  -  2/'- 

19.  2  a«  -  6  a^Z*^  _|_  ^^3  _  3  ^5^ 

20.  5  r^  -  2  rs^  + 10  r^s  -  4  s\ 

21.  20x^-5xy-\-4.xy-y^ 

22.  6ac-3G +  2  ad  — d. 

23.  8aic +155y  — 12a3/-10^^. 
Solution.    Sax  +  16  by  —  12  az^  —  10  &a:  = 

8  ax -12  aij  —  lObx -^l^by  = 
4  a  (2  a;  -  3  ?/)  -  5  &  (2  a;  -  3  ?/)  = 
(2a:-3?/)(4a- 56). 

24.  ac-bc-ad-\-  bd.  29.  2  a^  -  6  a%^  -ab^  +  3  b\ 

25.  ac  -  a^  -  Z»c  +  ^*^.  30.  10a^-25a%^  +  5b^-2ab. 

26.  2am-65m  — 3a7^  +  9&7^.  31.  l^x^  —  35ax'^  +  10a-A.x^, 

27.  37^ic-15A;ic-%  +  57c7/.  32.  30 a;^  -  10 cc^  + 1  _  3 ^^s. 

28.  x^-3xy-2xhf  +  ^y\  33.  a^**  -  2&«  -  5  a5 +10; 

34.  Solve  for  x,  x (a  -\- b)  =  r (a  -\- b) -^  s {a  -{-  b), 

35.  Solve  for  x,  mx  -\-  nx  =  vir  -\-  nr  -\-  ms  +  ns. 

Hint.  In  the  preceding  equation,  and  in  similar  ones,  the  value  of  the 
unknown  should  be  obtained  by  the  use  of  factoring.  Mental  division 
should  be  employed,  not  ordinary  long  division. 

36.  Solve  for  y,  ay  -{- by  =  ac  -\- be  —  ab  —  b'^. 

37.  Solve  for  z,  kz  —  Iz  =  hk  —  hi  —  k^  -\-  Ik. 

38.  Solve  for  y,  cy  -j-  ad  —  ae  =  dc  —  ce  -\-  ay. 

39.  Solve  for  m,2k  +  hl  —  mk  —  kl-\-2h  —  mh. 


FACTORING  121 

61.  Trinomials  which  are  perfect  squares.  Here  the  type 
form  is  a2d=2a&+&^ 

This,  by  page  105,  gives  us  the  two  expressions : 

a^-2ab  +  b^  =  (a-by. 

If  an  algebraic  expression  is  the  product  of  two  equal 
factors,  it  is  said  to  be  a  perfect  square. 

A  trinomial^  arranged  according  to  the  descending  powers 
of  one  letter,  is  a  perfect  square  if  the  first  and  third  terms 
are  positive  and  if  the  absolute  value  of  the  middle  term  is 
twice  the  product  of  the  absolute  values  of  the  square  roots  oj 
the  other  two  terms. 

Thus  in  the  type  form  above,  the  middle  term  2  aft  =  2  •  Va^  •  VP. 
Similarly,  the  trinomial  4  x^  —  20  xy^  +  25  ?/*  is  a  perfect  square, 
since  the  middle  term  20  xt/^  =  2  >  VJ^  •  V25y*  =  2  -  2  x  -  6  y\ 

ORAL  EXERCISES 

Form  perfect  trinomial  squares  by  supplying  the  missing 
terms  in  the  following : 

1.  ^2+  (?)  +  x\  11.  ^2  _  4^  _^  ^9). 

2.  c^+  (?)  +  25,  12.  x^-{- 10 X  +  (?). 

3.  52+  (?)  -f  16 1\  13.  c^-Sc-i-  (?). 

4.  A2+(?) -j-1,  U.  n^- 12  n -{-(?). 

5.  l  +  (?)-f-4A^  15.  n'-20n-\-(?), 

6.  9x^-(?)-^4:.  16.  x^-lSx-h(?y 

7.  c''-h2cd  +  (?).  17.  x^-  26 X  +  (?). 

8.  a^  -  2  ax -{- (?),  18.  4r2-4r  +  (?). 
9-  2/'-6  7/-f(?).  19.  9x^-\-6x-{-(?). 

10.y  +  2r  +  (?).  20.  9a2-6a  +  (?). 


122  FIEST  COUESE  IIST  ALGEBEA 

21.  16x^-24.x-{-(?),  24.  (?)  +  10r  +  25. 

22.  16x^-4.0x-\-(?),  25.  (?)-18^  +  81. 

23.  25x^-60x  +  (?).  26.  (?)  -  8  a^  +  x\ 

For  obtaining  one  of  the  two  equal  factors  of  a  perfect 
trinomial  square  we  have  the 

Rule,  Arrange  the  terms  of  the  trinomial  according  to  the 
descending  powers  of  some  letter  in  it, 

Extract  the  square  root  of  the  first  and  third  terms  and 
connect  the  results  hy  the  sign  of  the  middle  term. 

Before  applying  the  foregoing  rule  one  should  never 
forget  to  observe  whether  the  expression  to  be  factored 
is  a  perfect  trinomial  square  or  not. 

ORAL  EXERCISES 

Factor  the  following : 

1.  ic2+4a;  +  4.  13.  c'-\-2c-{-l, 

2.  Z,2_^6Zi  +  9.  U,l~2x-hx^ 

3.  c^-Sc-\-16.  15.  4aj2+4x  +  l. 

4.  ^2+ 10 a +  25.  16.  9^2+ 6c +  1. 

5.  d^-lOd-i-25.  17.  9^2+ 12c +  4. 

6.  -h^-Uh-hSe,  18.  16x2+ 8  a; +1. 

7.  49-14cc+a;l  19.  16 0^2-24 x  + 9. 

8.  64  +  16m  +  m2.  20.  25 x^  —  20 x -{- 4.. 

9.  81-18m  +  ml  21.  9  +  42  a;  +  49  a;^. 

10.  71^-20  71  +  100.  22.  16-24:X-\-9x\ 

11.  A;2_22;b  +  121.  23.  25-^0x-\-16x\ 

12.  144  -  247/  +  y\  24.  49  +  70aj  +  25x\ 


FACTOEING  123 

EXERCISES 

Write  the  factors  for  the  following : 

1.  9?^  +  6rs  +  sl  10.  121a'^-Uab  +  4:b^ 

2.  4.a^  +  4.ax-}-x\  11.  81  x^  + 126 xy  +  49 y\ 

3.  9x^-6xy-{-7/.  12.  Sex^  +  25f-60xy, 

4.  16x^-Sxm  +  m\  13.  169  a^-^9b^-7S  ah. 

5.  m^  + 10  m/z  +  25  ril  14.  49  d''  +  210  cd  +  225  c^. 

6.  2bif-10yx-\-x\  15.  196  a'^^  - 140  a^  +  25  ^»l 

7.  81a2+18a^»  +  ^''.  16.  9  a%^ -12  ah  ^4., 

8.  ?/i2  -  26  mTi  + 169  n\  17.  16  cV  +  56  ccc  +  49. 

9.  9  7^2  _  0Q  ^;^  _|.  jLOO  k\  18.  9  mV  -  24  mnp  +16^. 

It  is  only  in  the  beginning  of  factoring  that  polynomials 
are  classified  for  the  student.  In  the  practical  work  of 
handling  fractions  and  solving  equations  he  must  deter- 
mine for  himself  the  type  of  the  polynomial  to  be  factored. 
It  is  therefore  very  important  that  he  fix  in  mind  the 
various  types  and  the  manner  of  factoring  each.  More- 
over, he  should  remember  that  the  polynomials  which  arise 
in  practice  often  have  three  or  more  factors.  Miscellaneous 
review  exercises  afford  excellent  practice  in  recognizing 
types  and  in  determining  all  the  prime  factors. 

At  this  point  the  suggestions  given  on  page  135  will 
prove  helpful,  though  only  the  first  three  of  the  types  there 
given  have  as  yet  been  considered. 

REVIEW  EXERCISES 

Separate  into  prime  factors  : 

1.  x^  -f-  6x^  +  9a^.  5.  a^  -\-2a^x  +  ax". 

2.  2x^-\-4.x  +  2.  6.  2^^-20^2  +  506'. 

3.  x^-10x2_^25a:.  7.  lOOic- 80a^2_|_-|^6^8 

4.  a^-{-2a%  +  ab\  8.  98  c  +  28  c^  +  2c«. 


124  FIEST  COUESE  IN  ALGEBEA 

9.  80  r  -  40  7-2  +  5  rl  13.  126  cH^  +  147c^<^  +  21  cd\ 

10.  245a;^-140cc'  +  20ccl  14.  2  ac  +  4cx  +  2ay  +  4aj2/- 

11.  45 m^-  60 mTi  +  20 tiI  lb.  2ax  -  ^hx -\-2ay  -  6 ^?/. 

12.  2  aa;  +  4  ao?^  +  2  (xxl  16.  3mr  —  6  7ir+15ms  — 30/15. 

17.  12  (^5  —  6  ^c  —  6  aa;  +  3  cic. 

18.  Solve  for  x,  ax  -\-  ex  =  m  (a  -\-  c)  —  n(a  -\-  c). 
■   19.  Solve  for  x,  (a  +  b)x  =  a^  +  2ab  +  b\ 

20.  Solve  for  m,  mh  —  mk  —  }i?  —  2}ik-\-W'. 

21.  Solve  for  n^  nr  —  2 ns  =^  r^  —  ^rs  -\-  A, s^. 

22.  Solve  for  2/,  ky  -  4.kl  -  P  =  4.k^  -ky  -  ly. 

23.  Solve  for  3/,  acy  —  ac^^  —  ac^  =  ady  —  2  acd. 

24.  Solve  for  z,  2  ade  —  ae^  =  ad^  —  adz  +  aez. 

62.  A  binomial  the  difference  of  two  squares.  The  type 
form  is  a^  —  b^ 

By  page  107,        a^-h^=(a-\-b)(a-hy 

Hence  we  have  the 

Rule,  Regarding  each  term  of  the  binomial  as  positive^ 
extract  its  square  root. 

Add  the  two  square  roots  for  one  factor^  and  subtract  the 
second  from  the  first  for  the  other, 

EXABIPLES 

1.  Factor  c*  —  d'^. 

Solution,    c*  -  df*  =  (c^  +  d^)  (c^  —  d^),  by  application  of  the  rule ; 
=  (c^  +  d^)  (c  +  d)  (c  —  d)f  by  application  of  the 
rule  to  c^  —  d^. 

2.  Factor  Sla^-ldy"^.  , 
Solution.    81  a8  -  16  y^^  =  (9  a*  +  4  /)  (9  a*  -  4  y^) 

=  (9  a*  +  4  ?/6)(3  a2  +  2^/3)  (3  a^-  2  z^> 


FACTOEING 

12S 

Factor : 

ORAL  EXERCISES 

1.  c"  -  d\ 

8, 

40^2  _  25. 

15. 

169  a" 

-100^^2. 

2.  a^-A. 

9. 

9aj2_42/l 

16. 

25x^- 

•S6y\ 

3.  A^  - 1. 

10. 

16ic^-yl 

17. 

36x'- 

■49  2/1 

4.  4m2-l. 

11. 

9^2-64  6^1 

18. 

16x'  - 

121^2. 

5.  9-rl 

12. 

100  ^2  -  121  Z^2. 

19. 

64.x'  - 

■81^^ 

6.  1-A^l 

13. 

121aj2-49  2/'. 

20. 

1000^2 

-i2iy. 

7.  1-4^:^. 

14. 

144  c^- 121(^1 

21. 

144  x«- 

-1692/«. 

Solve  for  x^ 

22.  x(a  +  3) 

=  a' 

'  -  9.              24. 

x{k^-l)=k'' 

-49. 

23.  x(G-h^-. 

=  c2 

-  25.             25. 

x(2-^ 

■  c)  =  4  - 

-c\ 

EXERCISES 

Factor  the  following : 

1.  a*  -  b\  6.  81  -  c\  11.  c'd'  -  1. 

2.  x'-l.  7.  625  a:* -1.  12.  c«  -  81. 

3.  a^-16.  8.  625 -a^  13.  c' -  d\ 

4.  r*-81.  9.  625x'-16y\  14.  a^b'-W, 

5.  16  a*  -  1.  10.  a^''  -  c^^  15.  xy  -  z\ 

16.  (o^  +  yf  -  z\ 
Hint,    (x  +  y^  -  z^  =  [{x  ■\- y)  ^  z]  [(x  +  2/)  -  2]  etc. 

17.  (x  -  yf  -  25.  20.  4  (aj  -  9)2  -  9  a^. 

18.  (2  (T  +  5)2  -  4  /.  21.  9  (6^  -  ^)2  -  25  cl 

19.  (3a-7)2-(r2.  22.  25  x\c -\- df  -  9  y\ 
2f3.  4«^2_^2c-^)2. 

Hint.   4  a2  _  (2  c  -  (!)2  =  [2  a  +  (2  c  -  d)]  [2  a  -  (2  c  -  d)]  etc. 

24.  25a^2_(^_3^^^2^  26.  81  a^  -  4  (c*  +  2)^. 

25.  49  a^  -  (3  a;  -h  2)1  27.  100  -  a\b  +  (j)^. 


126  FIEST  COURSE  IN  ALGEBRA 

28.  121  a^-4.P(c-S dPf.  33.  25 a^'lP'  -  9 (a  +  h)\ 

29.  yV  -  (y  4-  ^)'.  34.  4  o2^2  _  (^  _  2  (^)2. 

30.  cV-(c-e)-^.  35.  16  c^d:^  -  (ci^  +  ^)2^ 

31.  c2^2  -  {(?  -  df.  36.  25  c^d:^  _  32^^  _|.  ^^^2^ 

32.  9a%^-4:(a~by.  37.  49  A^  -  9 (r^  -  s^)^. 

Some  polynomials  of  four  or  six  terms  may  be  arranged  as 
the  difference  of  two  squares  and  factored  as  in  the  preceding 
exercises. 

EXERCISES 

Factor: 

1.  x^  -^2x7/  +  ]f  —  z\. 

Solution.    x^  +  2xy  -^  y^-z^  = 
(x  +  yy-z^  = 
(x-\-  y  ■}■  z)(x  +  y-z). 

2.  x^  +  2x-hl-  if.  4.  c^  -  10c  +  25  -4.d\ 

3.  a^^6a  +  9-P.  5.   4^0" -^  4.a -{-1  -  9b\ 

6.  16  +  24a+9«^2_25^2^2^ 

7.  Aa'-12ab  +  9b^-9b\ 

8.  25x^  +  A2/-20xy-16z\ 

9.  9:^2  +  25i/  -Slz*-  SOxtf, 

10.  60ab-25c'-j-9b^-^100a\ 

11.  l-Uab''  +  ^9a%^-b\ 

12.  4-20  aZ^V  +  25  ^^^^V  _  4  ^s^ 

13.  9ir2_3()^^_l_25^2_-|^g^4^ 

14.  16ir^- 8iz^y +  y'-^^ 

15.  Z^  —  m^  +  2  mri  —  n\ 
Solution,    l^  —  m^  +  2  mn  —  n^  = 

P  —  (wP'  —  2  mn  +  w^)  = 

[Z  +  (m  -  w)]  [Z  -  (m  -  w)]  = 
(^t  +  m  —  n)  Q  —  m  ■\-  w). 


PACTORING  127 

16.  x'-4ty'-[-A:yz-'z\  18.  c"  -  4.0" -12 ah  -  ^b\ 

17.  x'-if-\- 10  ijz  -  25 z\  19.  ^c'-a'  +  lOab-  2W. 

20.  9c^2__^2^5^^_9^2^ 

21.  16 e"  -  25m^  +  10mn  -  n\ 

REVIEW  EXERCISES 

Factor : 

1.  ax'-af,  10.  5aV-45c^ 

2.  a^c  -  4  6^6.  11.  aV-  2  aV+  aa) 

3.  a«  +  6  a'b  +  9  a^>l  12.  ax^-  a^x 

4.  c«  +  2c^  +  c^.  13.  2x^-162cc. 

5.  dx''-  25  d\  14.  xhj  -  16a;y^. 

6.  25e*-30c^+9e^  15.  5a*-5. 

7.  2x-^-18V  16.  x^-xy\ 

8.  r^  +  2  r^l-  r^  17.  32  aj*-  1250. 

9.  a«^  -  6 ^2^*^+  9 a^^^  18.  :dz{x  +  yf-  12 ;5J«. 

19.  20ic2+20x«+5cc*+5x^ 

20.  8 r*- 8/^(35  + 0'- 

21.  8a^-12G^2^c-18a2^>2_2aV. 

22.  Solve  for  x,x{a  +  2)  =  a^—  4. 

23.  Solve  for  y^Vy  ^hy  =  h^-  25. 

24.  Solve  for  z,  z  (c^^  25)  (c  -  5)  =  c*-  625. 

25.  Solve  for  m,  mh'^—  mhk  =  A^—  AA:^. 

26.  Solve  for  n,  acn  -{-2 c  —  ah  =  2cn  —  2c, 

63.  The  quadratic  trinomial.    The  type  form  is 

For  many  trinomials  of  this  type  two  binomial  factors 
may  be  found  of  the  form  (x -\- r)  (x -\- s).  The  method 
of  factoring  to  be  used  is  the  reverse  of  the  method  of 


128  FIEST  COUESE  IN  ALGEBRA 

multiplying  two  binomials   given   on  page  109.    From  a 
study  of  the  four  examples  there  given  it  is  evident  that 

(1)  The  first  term  of  each  binomial  factor  is  the  square 
root  of  the  first  term  of  the  trinomial 

(2)  The  second  terms  of  the  binomials  are  those  factors 
of  the  last  term  of  the  trinomial  whose  algebraic  sum  equals 
the  coefficient  of  the  middle  term  of  the  trinomial. 

EXAMPLES 

1.  Factor  ic^  +  12  cc  +  32. 

Solution.  0:2  +  12  2:  +  32  =  (x  +  ?)  (a:  +  ?). 

It  is  necessary  to  find  two  numbers  whose  product  is  +  32  and 
whose  sum  is  +  12. 

Now  32  =  1 .  32  =  2  .  16  =  4  .  8. 

The  first  two  pairs  are  rejected,  for  each  fails  to  give  the  sum 
+  12.  The  third  pair  of  factors  of  32,  namely  4  and  8,  gives  the 
correct  sum. 

Therefore  x^  +  12  x  +  32  =  (x  +  8)  (ar  +  4). 

2.  Factor  a^— llc^  +  24. 

Solution.  Since  24  is  positive,  its  two  factors  must  have  the  same 
sign ;  since  — 11  is  negative,  both  factors  must  be  negative.  I^ow 
24  =  1  •  24  =  2  •  12  =  3  •  8  =  4  •  6.  By  inspection  of  these  products, 
—  3  and  —  8  are  found  to  be  the  required  numbers. 

Therefore  a^  -11  a  -{■  24:  =  (a  -  3)  {a  -  8). 

3.  Factor  c^  -  c  -  42. 

Solution.  The  product  —  42  is  negative  ;  hence  the  required  factors 
have  unlike  signs.  The  sum,  —  1,  being  negative,  the  negative  factor 
of  —  42  must  have  the  greater  absolute  value.  !N'ow  42  =  1  •  42  = 
2  .  21  =  3  .  14  =  6  .  7.   We  see  that  6  +  (-  7)  =  - 1. 

;  Therefore  c^  -  c  -  42  =  (c  +  6)(c  -  7). 


FACTORmG  .        129 

EXERCISES 

Factor  : 

1.  a^  +  5  a  +  6.         6.  Iv"  -  16  72,  +  15.  11.  s'  +  ^s-  35. 

2.  c2  +  7c  +  12.        7.  7i2_87i  +  16.  12.  s^-Gs  +  O. 

3.  x^  +  Ax-\-l.         8.  F-177C  +  30.  13.  f  +  t  -  4.2. 

4.  ic2_8x  +  12.       9.  m^-107n  +  25.  14.  x2  +  5aj-14 

5.  m2-13m  +  30.  10.  7^-5^-14.  15.  10^-39  +  ^^ 

16.  a^-4a2-5.  17.  a* -14^2 +  49. 

18.  -a;2^aj  +  20. 

Solution.    By  taking  out  the  factor  —  1,  we  may  factor  as  usual. 
-  x^  -{-  X  -{-  20  =-l(x^  -  X  -  20)  =-l(x  -  b)(x  +  4) 

=  (5-x)(x  +  ^). 
Note  that  (—1)  is  not  retained  as  a  factor  in  the  final  result. 

19.  -,x^-x  +  12, 

20.  -m2-8m  +  9. 

21.  -2a +  63 -a''. 

22.  20  -  m  -  m\ 
2Z.  24.-{-2n-n\  34.  A^ _  3  /j.  _  130  j^2^ 

24.  80  -  2  r  -  r".  35.  /c^  -  4  A:^  +  4  Zl 

25.  cc2-cc-72.  >        36.  r2  +  15n9-10052. 

26.  a^-a-llO.  37.  nhk  +  k''-36h\ 

27.  a2  +  12a  +  36.  38.  r^s^  _  3  ^^^f  _  40  2^2 

28.  c^- 50  + 5a  39.  a^^^  +  8 a^c  +  7 cl 

29.  A2  _  55  _|.  5  ;^^  4Q^  33^2  _  ^2^2  _  g  ^^^^ 

REVIEW  EXERCISES 

Factor  the  following : 

1.  x^-\-lx^  +  12x\  6.  16a^-4.ah\ 

2.  a*  -  12  a«  +  36  a".  7.  20  c^  -  5  c^*. 

3.  2^2  +  lOiT  -  48.  8.  bx^-  20xy\ 

4.  5c«  +  15c2-140c.  9.  rh -\- 10  r^s^ +  21  rs\ 

5.  3w^*  +  66m«  +  363m2.  10.  2h^k^  +  2hV-12W. 


30. 

_46--21  +  5l 

K^31. 

27  +  6aj-cc2^ 

32. 

ir^  +  2  ra;  -  3  r^. 

33. 

m^  +  2  m/j  —  9S 

130  FIRST  COUESE  IN  ALGEBEA 

11.  2r^-162r,  15.  5  r^5  -  40  rV  +  60  r^s^. 

12.  45  mV  -  20  mn\  16.  39  d'- 10  cd^  -  c^d\ 

13.  2ajV+10xy  +  12V-  17.  IS  x' +  7 kx' -  k^x\ 

14.  4  c*6Z  +  4  c«<i'  -  24  c^d^  18.  8  a^^c^  +  a%^  -  65  c«. 

19.  6  aA-  +  3  h^c  + 18  aAA;  +  9  chk. 

20.  30  m^T^r  — 10  mr  +  45  m^n  — 15  m. 

21.  2  a«  +  4  ^2^  +  2  aZ^2  _  2  ^c^. 

22.  3  c«  +  6  c^cZ  +  3  C6^2  _ -j^2  c. 

23.  4  i»^  —  4  ccy^  —  8  xyz  —  4  x'^l 

24.  Solve  for  x,  x(a  +  2)=  a^ -j- 5  a -{-  6. 

25.  Solve  for  y,  ym  —  y  =  m^  —  Am  +  S. 

26.  Solve  for  z,  rz-\-r  =  7^-\-5z  —  20. 

27.  Solve  for  r,  ar  +  3  ac  —  «^^  =  2  cr  +  2  c^. 

28.  Solve  for  6^,  2  as  +  a^  +  a*  =  6  a^  +  a^s. 

64.  The  general  quadratic  trinomial.    The  type  form  is 

For  many  trinomials  of  this  type,  two  binomial  factors  of 
the  form  (^hx  +  A)  (mx  +  n')  may  be  found.  The  method  of 
factoring  such  trinomials  is  illustrated  in  the  following: 

EXAMPLES 

1.  Factor  2x'^  -{-5x  +  3. 

Solution.   2  x2  +  5  a;  +  3  =  (?:r  +  ?)  (?:r  +  ?).  _>^>^              ^2) 

To  get  the  proper  factors  we  must  supply  2  x^  +  ?  a; 

such  numbers  for  the  interrogation  points  in  ^  9  x  -\-  ^ 

(1)  and  in  (2)  as  will  give  (3).  2x^  +ox  +  3      (3) 

2  x^  for  the  product  of  the  first  two  terms  of  the  binomials, 
-f  3  for  the  product  of  the  last  two  terms  of  the  binomials,  and 
+  5  a;  for  the  sum  of  the  cross  products. 
Now  2  x^  =  2  X  •  Xy 

and  +3  =  1.3. 


FACTORING 


131 


The  factors  1  and  3  may  be  substituted  for  the  interrogation 
points  in  (1)  and  (2)  in  either  of  the  following  ways: 

^  _^  g      (Incorrect)  ^  _^  ^      (Correct) 

The  first  pair  is  rejected,  for  it  fails  to  give  a  product  having  the 
required  middle  term,  +  5  a:.  The  second  pair  gives  the  correct 
product. 

Therefore  2  x^  +  o  x  +  3  =  (2  x -^  3)(x +  1). 

2.  Factor  3(;c2_^_iq^ 
Solution.     3  x^  =  3  X  •  X 

-10  =  l--10==-1.10  =  2.-5  =  -2.5. 

Test  the  following  pairs  of  binomials : 
3a:  +  l         a:  +  13:i:-l         a;-13a:  +  2       a:  +  2    3  2-2       x-2 
a:- 10    3a:- 10     x  +  lO    3a:  +  10     x-5    'd  x  -  5       x  +  5    3x  +  6 

Only  the  last  pair  gives  the  desired  product. 


Therefore 


3x2 


-  10  =  (a:  -  2)  (3  a:  +  5). 

After  a  little  practice  it  will  usually  be  found  unnecessary  to  write 
down  all  of  the  pairs  of  binomials  that  do  not  produce  the  required 
product. 

If  none  of  the  pairs  gives  the  required  product,  the  given  trino- 
mial is  prime. 

EXERCISES 


7.  lOx'  +  rSx-S. 

8.  6^2  +  7^-20. 

9.  10a;2  4-9a:  +  2. 

10.  6x^  +  5x-6, 

11.  2cc2  +  13a;  +  18. 

12.  3^^2-10  6^-25. 

13.  5x^-3Sx-16. 

14.  10x''-h7x-6. 
X5.  4^2+20^  +  21, 


Factor : 

1. 

3  a^  -{-7x 

+  2. 

2. 

2x^-^rlx 

+  6. 

3. 

^x'  +  ^x 

+  5. 

4. 

2a^-9a 

+  10. 

Hint.    Since    the   last 
positive  and  the  second 
only  negative  factors  of 
be  considered. 

term   is 

negative, 

10  need 

5. 

5a^-2a 

-3. 

6. 

&a^  +  7a 

--5. 

132 


FIEST  COUESE  IN  ALGEBEA 


16.  9x^  +  Sx-2. 

17.  16c^-Sg-S. 

18.  lOP-71-12. 

19.  12x^-Sx-15. 

20.  14r2-39r  +  10. 

21.  21x^-61x  -SO,' 
^22.  255^-155  +  2. 

23.  36x^-36x-{-5. 

24.  36^2  +  23^-3. 

25.  mx'-21x  +  2. 

26.  2'\-x-lbx\ 

27.  12x^-\-x^-20. 

Hint.  12x*-{-x^-  20  = 

12  (x2)2  +  a;2  -  20. 


28.  50x' +  5aj2-  3. 

29.  6  +  7a^2_5^4, 

30.  12  +  17i^^  +  6ic«. 

31.  a2-3a^  +  252. 

32.  w'-{-2ab-  SP. 

33.  c2-c6^_12  6Zl 

^  34.  2a^2  +  5i:cy  +  22/l 

35.  2a2-5a^»  +  2^2^ 

36.  Sx^-10x7/  +  3y\ 

37.  10a^2-27cc2/  +  5  2/'. 

38.  12x^-\-2Sxi/-2y\ 

39.  30cc2-13iry-y2. 

40.  30cc2_|_j^()9^^_^3Q^2 


5.  27a;^-36;z;*  +  12aj^ 

6.  8a^^  +  2a;2_;j^^ 

7.  60x^-S5x^-60x. 

8.  3;:cV-4xy +  ict/^ 


REVIEW  EXERCISES 
Factor  the  following : 

1.  4:X^-hlOx  +  4.. 

2.  3aj«  +  18ic2_|_27^, 

3.  20cc*-60^^  +  45a;l 

4.  50  aa;2  -  140  a:«  +  98  ^. 

9.  12icV  +  21iry  ~6iC2/^. 

10.  5x^y-^10xy-75xy\ 

11.  6 xV  +  12  ar'x  +  4.rx^-{~S  axr. 

12.  —  18  ax'^y  —  45  a^ic^/  —  6  acc^  —  15  a^x. 

13.  45xy +  21a:V- 6cc'. 

14.  Solve  for  r,  r (a  +  1)  =  2  a"  +  3  a  +  1. 

15.  Solve  for  s,  sh  ^  sc  =  h^ -he  -  2  c". 

16.  Solve  for  ir,  2hx  =  Q>h^  +  h —  2 -\- x, 

17.  Solve  iov  y,  cy  —  cd  —  2  (^  =  dy  —  3  d^. 


FACTORING  133 

65.  A  binomial  the  sum  or  the  difference  of  two  cubes. 

The  type  form  is  n^-A-h^ 

a^+b^  divided  by  (a +  5)  gives  the  quotient  a^  —  ab  -\-  b^, 
and  a?  —  b^  divided  by  (a  —  b^  gives  the  quotient  a^-\-ab-{-V^. 

Therefore     a^  +  b^=^(a'\'  b)  (a^  -- ab  +  b^),  (1) 

and  a^-^b^  =  (a--  b)  (a^  +  ab  +  b^).  (2) 

Formulas  (1)  and  (2)  above  may  be  expressed  in  words 
as  follows: 

(1)  The  factors  of  the  sum  of  the  cubes  of  two  terms  are 
(a)  a  binomial  which  is  the  sum  of  the  terms  and  (Ji)  a 
trinomial  in  which  the  terms  are  the  square  of  the  first  term 
minus  the  product  of  the  first  and  the  second  term  plus  the 
square  of  the  second  term. 

(2)  The  factors  of  the  difference  of  the  cubes  of  two  terms 
are  (a)  a  binomial  which  is  the  difference  of  the  terms  and 
(V)  a  trinomial  in  which  the  terms  are  the  square  of  the  first 
term  plus  the  product  of  the  first  and  the  second  term  plus  the 
square  of  the  second  term. 


EXAMPLES 

1.  Factor  a^  +  27. 

Solution.    a3  +  27  =  aS  +  33  =  (a  +  3)  (a2  -  a  .  3  +  S^) 
=  (a  +  3)(a2-3a  +  9). 

2.  Factor  27  a^- 64  Z*^. 

Solution.    27  a^  -  64  6^  =  (3  a^y  _  (4  ly 

=  (3  a2  -  4  6)  [(3  a2)2  +  (3  «2)  (4  j)  +  (4  j^S] 
=  (3  a2  _  4  5)  (9  a4  ^  12  a%  +  16  h% 


134  FIRST  COUESE  IN  ALGEBRA 

EXERCISES 

Eactor  the  following : 

1.  x^-\-y\  7.  %-x\  13.  Sa^y  +  l. 

2.  m^-n^.  8.  64  +  2/'-  14.  216  -  a;l 

3.  a^  +  8.  9.  27  +  8  al  15.  m^  +  n\ 

4.  h^  -  64.  10.  m^  -  7^1  16.  x""  +  y^ 

5.  c^  +  8  d\  11.  125  cc^  -  f,  17.  x^  +  2/''- 

6.  7?i^  -  125  Til  12.  1-27  mV.  18.  64  x^  +  ^^ 

19.  a^  +  64.  20.  m^-n\ 

Hint.  This  expression  may  be  regarded  either  as  the  difference  of  two 
cubes,  (m^)^  —  (n^)^,  or  as  the  difference  of  two  squares,  (m^)^  —  (n^)'^. 
Since  the  difference  of  two  squares  is  one  of  the  simplest  type  forms  to 
factor,  one  should  always  use  it  when  possible. 

Thus  m^  —  n^  =  (m^  +  77,^)  {m^  —  n^)  etc. 

21.  x^  -  y\       23.  x'  -  64  y\      25.   64  x'  -  1.      27.  x^''  -  1. 

22.  a'  -  64.      24.   1  -  a'.  26.  a^^  -  h\         28.  y^''-  x\ 

REVIEW  EXERCISES 

1.  x'  +  xif.  5.  x^-2x^-\-21x-M. 

2.  2x\j-2  xy\  6.  x^-lx^  -  8. 

3.  2  a* -54  a.  7.  o^'' -  64  x^ 

4.  5x^-40x1  8.  3aj*  +  2x^~24ir-16. 

9.  Solve  f or  cc,  3  rr  —  aoj  =  27  —  a^ 

10.  Solve  for  x,hx  —  ^  =  h^  —  2x. 

11.  Solve  for  x,  c'x  -  27  =  c^ -j- 3cx  -  9x. 

12.  Solve  for  r,  a^  -\-  a^r  =  S  —  2  ar  —  Ar. 

13.  Solve  for  s,  abs  -^  as  -\-  a  =  ab^  —  obH, 


FACTOEING  135 

66.  General  directions  for  factoring.  Since  no  general 
method  of  factoring  can  be  stated  in  a  few  simple  rules, 
the  process  must  be  learned  by  means  of  such  type  forms 
and  typical  solutions  as  are  given  in  the  precedmg  pages. 
When  once  these  have  been  thoroughly  mastered,  readi- 
ness in  factoring  expressions  which  are  represented  by 
them  becomes  a  matter  of  experience.  Usually  a  student 
finds  it  comparatively  easy  to  factor  a  list  of  exercises 
classified  under  a  particular  type  form,  yet  a  list  of  mis- 
cellaneous exercises  he  finds  difficult.  This  usually  indi- 
cates inabihty  to  determine  the  type  of  an  expression  from 
its  appearance.  Until  the  student,  by  careful  study  of  the 
type  forms,  has  acquired  the  ability  to  do  this,  he  will 
make  little  progress.  There  are  many  types  that  are  not 
included  in  this  book,  which  the  student  who  continues 
the  study  of  algebra  will  meet  later. 

The  following  suggestions  will  prove  helpful  in  solving 
the  types  here  considered : 

7.  First  look  for  a  common  monomial  factor^  and  if 
there  is  one  (other  than  i),  separate  the  expression  into 
its  greatest  monomial  factor  and  the  corresponding  poly- 
nomial factor. 

II.  Then  hy  the  form  of  the  polynomial  factor  determine 
with  which  of  the  following  types  it  should  he  classed  and 
use  the  methods  of  factoring  applicable  to  that  type. 

1.  ax-\-ay-\-hx-\-hy.  4.  x^-{-hx-{-c. 

2.  0^=1:2  a&-h  ft".  5.  ax" -\- bx -{- c. 

3.  (^-V.  6.  a^:hb\ 

m.  Proceed  again  as  in  II  with  each  polynomial  factor 
obtained  until  the  original  expression  has  been  separated  into 
its  prime  factors. 

BE 


136  FIEST  COURSE  IN  ALGEBEA 

'      MISCELLANEOUS  EXERCISES 

Factor  : 

1.  x^-4.x.  IS.  S-{-15x-2ax-  10  ax\ 

2.  2  6^3  +  4  a%  +  2  ab\  19.  x^  -  27  a^^  -  x^  +  27. 

3.  aaj^+acc^y +  «^V+a2iC2/.  20.  a^-}-Sa%-aP-3b\ 

4.  4 i«*+  400 x'-  116 x«.  21.  20 x^y  +  38 c^y  -  30 xy. 

5.  2  cx^  +  2  caj' -  12  caj.  22.  S  ax'' -  2  a  -  20 x"^ -\- 5. 

6.  5  a«  -  10  a^^  -  75  a&l  23.  9a2  + 4^»2  _l2a5-4ccy. 

7.  lOcc*- 5ic^-30icl  24.  4 x^ -  icy -  4 a^^/ +  ^z'^^. 

8.  5  aj^'  +  40  a;l  25.  ir^  +  y2  _  4  ^2  _  2  ^^. 

9.  xV  + 3x^  +  8?/ +  24.  26.  x^ -  4:xY  +  ^Y  -  4.y^. 

10.  i«^  -  4 ic^  -  8 a^2  _|_  32.  27.  i»2+  ic^-  6 a:^ 

11.  16cc*- 8x^-2^  +  1.  28.  l-x^-^x^-x\ 

12.  »"  -  25  «^^  -  ^2  +  25.  29.  1-4  x^-{-  8  a^^-  32  x'. 

13.  -  45  aV  -  3 a'x^  +  18 a^cc.  30.  x^-  Sx^-^  16  x\ 

14.  9x^-S0axy''  -j-25ay.  31.  2i;cV+4ir2y  +  2a^7/- 2cc2/' 

15.  aj^  -  63  cc«  -  64.  32.  8  a^  -  8  ay24-  2  y»-  2  y. 

16.  cc^-64  2/^.  33.  4x*+4cc^-4;r«+irl 
17^  60-.*  -  7 x^y  -  5 3/'.  34.  a^-  13 a^+  36 c^l 

35.  xY  -  4  ^y  +  3  i»V'  - 12  xy. 

36.  Solve  for  x,  «.x  +  ac?  +  5c?  =  ac  +  5c  —  5x. 

37.  Solve  for  x,  ax  -\-  bx  +  S  =  a  -\-  b  -\-  S  x, 

38.  Solve  for  x,  10  r  —  25  —  5  x  =  7*^  —  rx. 

39.  Solve  for  x,  4  c^^  _^  2  c^x  -  c"  =  -  ex. 

40.  Solve  for  ^,  47-2  +  9  +  3^  =  2r^  +  12r. 

41.  Solve  for  x,  m  +  42  —  7  x  =  m^  —  mx. 

42.  Solve  for  X,  a  +  15  +  5  x  =  2  a^  —  2  ax. 

43.  Solve  for  X,  -  3  c  -  X  =  2  c'  -  2  ex  -  2. 

44.  Solve  for  x,  6  7^  —  12  r  +  4 x  =  r^  —  r^x  +  4  rx  —  8. 

45.  Solve  for  y,  a^  —  c^  =  ay  —  cy. 


CHAPTER  XIV 

SOLUTION  OF  EQUATIONS  BY  FACTORING 

67.  Simple  equations.   A  simple  or  linear  equation  in  one 

unknown  is  one  which  may  be  put  in  such  a  form  that 

(a)  the  unknown  does  not  appear  in  any  denominator; 
(6)  only  the  first  power  of  the  unknown  is  involved. 

Thus  3a;  —  4=7,  2n  +  5  =  4n  +  l,  ax  +  b  =  0,  are  simple  equa- 
tions, (x  —  l)(x  +  ^)  =  (x-\-  4)  (x  —  6)  is  also  a  simple  equation, 
since  on  multiplying  out  it  .becomes  x^  +  2x  —  S—  x^  —  2x  —  24, 
from  which,  after  transposition,  we  get  4  a;  +  21  =  0. 

In  the  preceding  chapters  only  simple  equations  have 
been  considered. 

68.  Quadratic  equations.  A  quadratic  equation  in  one  un- 
known is  one  that  may  be  put  in  such  a  form  that 

(a)  the  unknown  does  not  appear  in  any  denominator ; 
(J)  the  second  but  no  higher  power  of  the  unknown 
is  involved. 

Thus  a;2  +  4  X  -  5  =  0,  3  a;^  -  4  =  5  x  +  6,  ax^ -\- hx  -\-  c  =  0,  are 
quadratic  equations. 

A  quadratic  equation  is  often  called  an  equation  of  the 
second  degree. 

The  term  in  a  quadratic  equation  which  does  not  involve 
the  unknown  is  called  the  constant  term. 

69.  Solution  of  equations.  The  methods  of  factoring 
given  in  Chapter  XIII  enable  us  to  solve  many  quadratic 

137 


138  FIEST  COUESE  IN  ALGEBEA 

equations.  In  the  solution  of  equations  by  factoring,  use 
is  made  of  the  following 

Principle,    If  the  product  of  two  or  more  factors  is  zero^ 
one  of  the  factors  must  he  zero,   . 

Two  or  more,  or  even  all,  of  the  factors  ma^/  be  zero,  but 
the  vanishing  of  one  is  sufficient  to  make  the  product  zero. 
Consider  the  equation,  in  factored  form, 

(x-2}(x-i}=0,  (1) 

If  this  equation  is  to  be  solved,  all  of  the  numbers  which 
satisfy  it  must  be  found.  That  is,  we  must  find  every 
value  of  X  for  which  the  product  on  the  left  of  (1)  is  zero. 
If  the  product  is  to  equal  zero,  the  foregoing  principle  re- 
quires that  one  of  the  factors  be. zero.  Hence  any  value  of 
X  which  satisfies  (1)  must  make  either  a;—  2  =  0  or  x—  4  =  0. 
Hence  x  must  equal  either  2  or  4.  On  substituting  2  for  x  in 
(1)  we  obtain  (2  -  2)  (2  -  4)  =  0,  or  0  .  (-  2)  =  0.  Hence  2 
is  a  root  of  (1).  On  substituting  4  for  a;  in  (1)  we  obtain 
(4  _  2)(4  -  4)=  0,  or  2  .  0  =  0.  Hence  4  is  a  root  of  (1). 
A  moment's  inspection  makes  it  clear  that  2  and  4  are  the 
only  roots  of  the  equation. 

ORAL  EXERCISES 

For  what  value  of  x  is  each  of  the  following  expressions 
equal  to  zero  ? 

1.  ir-2.  Z,  x-S.  5.  3ic-12. 

2.  x  +  5.  4.  x-f-10.  6.  2x  +  4:. 

7.  What  is  the  value  of  2  x  0?   of  (-  2)  x  0?  of  9  x  0? 
of  0x21? 

8.  Make  a  general  statement  which  includes  all  the  results 
of  Exercise  7. 


SOLUTION  OF  EQUATIONS  BY  FACTORING   139 

What  is  the  value  of : 

9.  (x  -1)(X'-  2)  when  x  =  S?  1?  0? 

10.  (x  ~  4)  (cc  -  3)  when  x  =  S?  2?  4? 

11.  (x  -S)(x-  5)  when  x=-l?  3?  6? 

12.  (2a^+l)(3cc-l)  whencc=-l?  ^?  0? 

13.  x{x  -  1)  when  x  =  0?  1?  2? 

14.  (x  -l)(x-  2)  (x  -  3)  when  x  =  l?  2  ?  3  ? 

15.  Sx(x  +  2)  when  ic=-2?  0?  2? 

In  Exercises  16-22 ^  which  of  the  numbers  at  the  right  of 
each  equation  is  a  root  of  that  equation  ? 

16.  (x-3)(x-l)=0.  1,  2,3. 

17.  (x  +  2)(x  -  4)=  0.  -4,-2,  2,  4. 

18.  .t(3cc-2)=0.  0,  1,  2. 

19.  (2x  +  l)(aj  +  3)=  0.  -  1   -2,-3. 

20.  (ic  -  1) (^  -  2) (x  -  3)=  0.  1,  2,  3,  4. 

21.  cc(aj  +  l)(^-l)=0.  -1,  0,  1,  2. 

22.  (x  -  iy(x  -  9)  =  0.  1,  9,  10. 

What  are  the  roots  of  the  equations  in  Exercises  23-28  ? 

23.  (x-l)(x  +  l)=0,  26.  (2a;-3)(ir-l)=0. 

24.  (x  +  2)(x-S)=0.  27.  (5x -i- 2)(x  +  2)=0, 

25.  (x-2)(aj-l)(cc  +  2)  =  0.   28.  (2^^  -  1)  (3aj  +  5)  =  0. 
29.  Is  there  any  one  value  of  x  which  makes  both  factors 

of  (x  —  S)(x  -\-  6)  equal  to  zero  ? 

EXAMPLES 

1.  Solve  the  quadratic  equation  x^  -\-  5  x  =  6, 
Solution.    Transposing,     x^  -[■  5  x  —  Q  =  0. 
Factoring,  (x  -  1)  (a;  +  6)  =  0. 

The  value  of  x  which  makes  the  factor  x  —  1  equal  to  zero  is  a 
root  of  the  quadratic.    Setting  x  —  1  =  0,  we  obtain  x  =1, 


140  FIEST  COUESE  IN  ALGEBEA 

Similarly,  the  value  of  x  which  makes  x  +  6  equal  to  zero  is  a 
root  of  the  quadratic.    Setting  a:  +  6  =  0,  we  obtain  a:  =  —  6. 

Hence  1  and  —  6  are  the  roots  of  the  given  quadratic  equation. 

Check.    Substituting  1  for  a:  in  a;^  +  5  a:  =  6,  we  have  1  +  5  =  6. 
Substituting  —  6  for  a:  in  a;^  +  5  a:  =  6,  we  have  36  —  30  =  6. 

2.   Solve  the  quadratic  equation  cc^  =  4cc. 
Solution.    Transposing,    x^  --  4  a:  =  0. 
Factoring,  a:  (a:  —  4)  =  0. 

The  factors  are  x  and  a:  —  4.  The  value  which  makes  the  first 
factor  zero  is  a;  =  0.  The  value  of  x  which  makes  the  second  factor 
zero  is  a;  =  4.    Hence  the  roots  of  ar^  —  4  a;  =  0  are  0  and  4. 

Check.    Substituting  a;  =  0  in  a;^  =  4  a:,  0  =  0. 
Substituting  a:  =  4  ina:2  =  4a;,  16  =16. 

For  solving  an  equation  in  one  unknown  by  factoring 
we  have  the 

Rule.  Transpose  the  terms  so  that  the  right  member  is  zero. 
Then  factor  the  expression  on  the  left^  set  each  factor  which 
contains  the  unknown  equal  to  zero,  and  solve  the  resulting 
equations. 

It  must  be  kept  in  mind  that  a  root  of  an  equation  is 
a  number  which  satisfies  the  equation. 

One  should  never  divide  each  member  of  an  equation  by  an 
expression  containing  the  unknown^  for  in  this  manner  roots 
may  be  lost 

Thus,  if  in  Example  2  we  had  divided  both  sides  of  the 
equation  by  x,  the  resulting  equation  ^ould  have  been 
a;  —  4  =  0,  which,  to  be  sure,  gives  us  one  root  of  the 
given  equation.  But  we  have  lost  the  root  x=0  which 
corresponds  to  the  factor  by  which  we  divided. 


SOLUTION  OF  EQUATIONS  BY  FACTOEING    141 

EXERCISES 

Find  the  roots  of  the  following  quadratic  equations,  and 
check  as  directed  by  the  teacher : 

1.  0^2  _  9  ^  Q  7^  x^  =  5x.  13.  x^-i-S=-6x. 

2.x^-25  =  0.       8.  3aj2  =  9^.  U.x^^  =  3x-j-10. 

3.  x^=16,  9.  x^-4.x  +  3  =  0.  15.  4.x^-16x  =  0. 

4.  x^  =  4.9,  10.  x^-6x-}-9  =  0.  16.  5 ic^  _^  35 i«  -  0. 

5.  x^-'2x  =  0.  11.  x2-7a;=-12.  17.  8-9cc  =  -ir2 

6.  2x^-{-6x  =  0.  12.  iK'  +  x  =  20.  18.  12ic-28  =  -x2. 
19.  i»'  -16x  +  64  =  0.                 30.  x^  =  4.P. 

^^,^0.  cc2  -  54  =  15  X.  31.  i«2  =  16  k^. 

21.  12.^  2gcc  +12^^  =  0.  32.  x'-2bx-j-b'  =  0. 

22.  3cc^  +  .i«  =  4.  ZZ.  x'-\-4.a'  =  4.ax, 

23.  18a;2=9r«  +  20.  34.  ic^  _  ^^  ^  0. 

24.  (^  +  8)(i:c+l)  =  -12.  35.  x"  =1  ax. 

25.  x:'  +  9x-12  =  3x-\-lh.  36.  2^5^  _  ^^  ^  21- 6a;. 
^26.  a;  +12  =  xi  37.  cc^  -  ^ic  -  ax  =  0. 

27.  50x  +  24  =  25a;l  38.  x^-^x  =  2ax. 

28.  (x-ll)(i:c  +  3)  =  2cc  +  42.     39.  x'^  -  ax  -  hx -{- ah  =  0. 

29.  x^  —  a'^=:  0.  40.  ic^  +  ^^cc  =  4 ^>  -|-  4a;. 

70.  Cubic  equations.     An  equation  in  x  which  may  be 
put  in  the  form 

aa?  +  hoiP'  -{-  ex  -\-  d  =  Q^ 

where  the  coefficients  a,  J,  e^  and  d  represent  numbers,  is 
called  a  cubic  equation,  or  an  equation  of  the  third  degree. 
Some  equations  of  higher  degree  than  the  second  may 
be  solved  by  the  method  of  factoring. 


142  FIEST  COURSE  IN  ALGEBRA 

EXAMPLE 

Solve  the  cubic  equation  x^  —  9  x  =  9  —  x\  (1) 

Solution.    Transposing,  x^  +  x'^  —  9  x  —  Q  =  0. 

Grouping,  x^  (x  +  1)  -  9  (r  +  1)  =  0. 

Factoring,  (x  +  I)  (x^  -  9)  =  0, 

or  (x  +  1)  (x  +  3)  {x-3)  =  0. 

Setting  each  factor  equal  to  zero, 

X  +  1  =  0,  whence  x  =  —  1, 

ar  +  3  =  0,  whence  x  =—  S, 

x  —  3  =  Of  whence  x  =  3. 

Therefore    —  1,   —  3,    and    3    are    the    roots    of    the    equation 
x^-9x  =  9-x\ 

Check.  When  a:  =  —  1,  (1)  becomes       —1  +  9  =  9—1. 

When  x  =  -3f  (1)  becomes  -  27  +  27  =  9  -  9. 

When  X  =  3,      (1)  becomes      27-27  =  9-9. 

EXERCISES 
Find  the  roots  of  the  following  equations,  and  check : 

1.  a;8-9x  =  0.  8.  3:^8- 2(r2 -12:^4- 8  =  0. 

2.  3x»  =  12a^.  9.  x^--25  =  25x-x\ 

3.  x^-i-4:x''-12x  =  0.  10.  2(x^-x)  =  3(l-x^y 

4.  a^-14:x  =  5x'',  11.  x^- 5x^  +  4:  =  0. 

5.  5x^  =  2x-3x^  12.  ic^-13(:c2  +  36  =  0. 
e.  a^-x^  -4.x-h4.  =  0.         IZ,  x^  =  4.x\ 

7.  2x^-x^  =  Sx-  4.  14.  x^  =  26cc2  _  25. 

PROBLEMS 

1.  The  square  of  a  certain  number,  plus  the  number  itself, 
is  42.    Find  the  number. 

Hint,   n^  -}-  n  =  42. 

2.  If  from  the  square  of  a  certain  number  three  times  the 
number  be  taken,  the  remaiuder  will  be  54.  Find  the  number. 


SOLUTION  OF  EQUATIONS  BY  FACTORING   143 

3.  If  to  the  square  of  a  certain  number  the  sum  of  twice 
the  number  and  9  be  added,  the  result  will  be  129.  Find  the 
number. 

4.  Three  times  the  square  of  a  certain  number  is  equal  to 
four  times  the  number.   What  is  the  number? 

5.  A  certain  number  is  added  to  16,  and  the  same  number 
is  also  added  to  21;  the  product  of  the  two  sums  is  546.  What 
is  the  number  ? 

6.  A  certain  number  is  subtracted  from  15,  and  the  same 
number  is  also  subtracted  from  25 ;  the  product  of  the  re- 
mainders is  119.    Find  the  number. 

7.  From  30  a  certain  number  is  subtracted,  and  the  same 
number  is  added  to  18  ;  the  product  of  the  results  thus  obtained 
is  560.    Find  the  number. 

8.  If  a  certain  number  be  added  to  15,  and  the  same  number 
be  subtracted  from  22,  the  product  of  the  sum  and  difference 
thus  obtained  will  be  36  more  than  50  times  the  number.  Find 
the  number. 

9.  If  from  the  square  of  four  times  a  certain  number,  five 
times  the  number  be  taken,  the  result  will  be  15  times  the 
square  of  the  number.    Find  the  number. 

The  student  has  probably  observed  that  a  quadratic  equation  has  two 
roots  —  one  corresponding  to  each  factor  which  contains  an  unknown. 
In  Problems  1-9,  where  the  question  is  asked  about  a  "  certain  number," 
both  roots  of  the  equation  are  solutions  of  the  problem,  since  both  are 
numbers.  In  problems  like  the  10th,  where  the  question  is  asked  about 
some  measurement,  it  frequently  happens  that  one  of  the  roots  is  a 
number  which  cannot  measure  the  particular  kind  of  thing  with  which 
the  problem  deals.  Thus  one  root  of  the  equation  to  which  Problem  10 
leads  is  —  20  ;  and  —  20  as  the  length  of  a  lot  is  meaningless.  Hence  we 
reject  it  as  a  solution  of  the.  problem,  in  spite  of  the  fact  that  it  occurs  as 
a  root  of  the  equation.  Similarly,  if  a  problem  dealing  with  dimensions 
yields  an  equation  with  a  negative  root,  or  if  a  problem  asking  '^how 
many  men"  yields  an  equation  with  a  fractional  root,  we  reject  these 


144  FIEST  COUESE  IN  ALGEBEA 

roots.  Although  they  satisfy  the  algebraic  conditions  of  the  equation 
to  which  the  problem  leads,  they  fail  to  comply  with  the  physical 
conditions  of  the  problem  itself,  and  consequently  should  not  be  re- 
tained as  answers. 

10.  The  depth  of  a  certain  lot  whose  area  is  1600  square 
feet  is  four  times  its  frontage.    Find  its  dimensions. 

11.  The  area  of  the  floor  of  a  certain  room  is  54  square 
yards.  The  length  is  3  yards  more  than  the  breadth.  What 
are  the  dimensions  of  the  floor  ? 

12.  The  area  of  a  rectangular  field  is  216  square  rods. 
The  field  is  6  rods  longer  than  it  is  wide.  Find  its 
dimensions. 

13.  The  sum  of  the  squares  of  two  consecutive  integers  is 
313.    Find  the  numbers. 

14.  The  sum  of  the  squares  of  two  consecutive  odd  integers 
is  514.    Find  the  numbers. 

15.  The  sum  of  the  squares  of  three  consecutive  odd  integers 
is  251.    Find  the  numbers. 

16.  An  uncovered  box  6  inches  deep,  with  square  bottom, 
has  112  square  inches  of  inside  surface.  Find  the  other  inside 
dimensions. 

17.  Eemembering  that  the  faces  of  a  cube  are  squares,  find 
the  edge  of  a  cubical  box  whose  entire  outer  surface  is  216 
square  inches. 

18.  A  rectangular  box  is  three  times  as  long  and  twice  as 
wide  as  it  is  deep.  There  are  550  square  inches  in  its  entire 
outer  surface.    Find  its  dimensions. 

19.  A  box  is  3  inches  longer  and  1  inch  wider  than  it  is 
deep.  There  are  62  square  inches  in  its  entire  outer  surface. 
Find  its  dimensions. 


SOLUTION  OF  EQUATIONS  BY  FACTOEING  145 

The  altitude  of  a  triangle  is  the  perpendicular  from  any 
vertex  to  the  side  opposite.    This  side  is  called  the  base. 

In  the  adjacent  figures  BD  is  the  altitude  and  ^C  is  the 
base  of  each  triangle. 

B  B 


If  a  is  the  altitude  of  a  triangle  and  b  its  base,  the  area  of 
the  triangle  is  -—  • 

In  making  use  of  this  and  similar  formulas  the  unit  in  terms 
of  which  the  lines  are  measured  must  be  specified. 

20.  The  area  of  a  triangle  is  40  square  feet ;  its  altitude  is 
8  feet.    Find  the  base. 

21.  The  altitude  of  a  triangle  is  twice  the  base  and  the  area 
is  36  square  feet.    Find  the  base  and  the  altitude. 

22.  The  base  of  a  triangle  is  5  times  the  altitude  and  the 
area  is  40  square  feet.    Find  the  base  and  the  altitude. 

23.  The  area  of  a  triangle  is  48  square  inches ;  the  base  is 
six  times  the  altitude.    Find  the  altitude  and  the  base, 

24.  The  area  of  a  triangle  is  24  square  feet ;  the  altitude  is 
2  feet  longer  than  the  base.    Find  the  altitude  and  the  base^ 

Hint.   Let  x  =  the  base  in  feet. 

Then  x  +  2  =  the  altitude  in  feet, 

,  x(x  +  2)      x2+2x      ^. 

and  — ^^ = =  the  area. 

2  2 

Therefore  ^        =  24. 

2 

Multiplying  each  member  by  2,  this  equation  becomes 
x^+2x  =  48. 


146  FIEST  COUliSE  IN  ALGEBRA 

25.  The  altitude  of  a  triangle  is  4  feet  longer  than  the  base 
The  area  is  6  square  feet.    Find  the  base  and  the  altitude. 

26.  One  leg  of  a  right  triangle  is  4  yards  longer  than  the 
other  and  the  area  is  30  square  yards.    Find  the  legs. 

27.  The  area  of  a  right  triangle  is  26  square  yards  and  one 
leg  is  8  feet  longer  than  the  other.    Find  the  legs. 

28.  The  area  of  a  triangle  is  2|-  square  feet  and  the  base 
is  6  inches  longer  than  twice  the  altitude.  Find  the  base  and 
altitude. 

29.  The  area  of  a  triangle  is  12  square  yards  and  the  alti- 
tude is  6  feet  less  than  twice  the  base.  Find  the  base  and  the 
altitude. 

A  trapezoid  is  a  four-sided  figure,  two  of  whose  sides  are 

unequal  and  parallel.  . — , —  ^ 

The  bases  of  a  trapezoid  are  the  /         | 

two  parallel  sides,  h  and  c,  /            f^ 

The  altitude,  a,  is  the  perpen- 


dicular distance  between  the  bases.  ^      /t  .     \ 

a(h'\-c) 

The  area  of  a  trapezoid  is  given  by  the  formula » 

30.  Find  the  area  of  a  trapezoid  whose  bases  are  10  and  18 
and  whose  altitude  is  12. 

31.  The  altitude  of  a  trapezoid  is  8  inches,  its  area  is  96 
square  inches,  and  one  base  is  4  inches  longer  than  the  other. 
Find  each  base. 

Hint.   Let  x  =  the  length  of  one  base  in  inches. 

Then  x  ■}■  4  =  the  length  of  the  other  base  in  inches, 

Six  4-  X  4-  4t) 
and  — ^ --',  or  8  X  -1-  16  =  area  of  the  trapezoid. 

Therefore  8x  +  16  =  96. 

32.  One  base  of  a  trapezoid  is  12  feet,  the  other  base  is  three 
times  the  altitude,  and  the  area  is  90  square  feet.  Find  the 
altitude. 


SOLUTION  OF  EQUATIONS  BY  FACTOKmG  147 

33.  The  altitude  of  a  trapezoid  is  one  half  the  shorter  base 
and  the  latter  is  two  thirds  of  the  other  base.  The  area  is  250 
square  feet.    Find  the  bases  and  the  altitude. 

34.  One  base  of  a  trapezoid  is  12  feet  longer  than  the  alti- 
tude, the  other  base  is  6  feet  longer  than  the  altitude,  and  the 
area  is  112  square  feet.    Find  the  bases  and  the  altitude. 

35.  The  bases  of  a  trapezoid  are  respectively  1  foot  and 
5  feet  longer  than  the  altitude,  and  the  area  is  30  square 
yards.    Find  the  bases  and  the  altitude. 

36.  One  base  of  a  trapezoid  is  6  feet  longer  than  the  other, 
the  altitude  is  one  half  the  sum  of  the  bases,  and  the  area  is 
9  square  yards.    Find  the  bases  and  the  altitude. 

37.  The  area  of  a  trapezoid  is  8  square  yards,  the  altitude 
equals  one  base,  and  the  other  base  exceeds  the  altitude  by 
2  feet.    Find  the  bases  and  the  altitude. 

38.  One  base  of  a  trapezoid  exceeds  the  other  by  10  feet, 
the  altitude  is  2  feet  longer  than  five  times  the  shorter  base, 
and  the  area  is  22  square  yards.  Find  the  altitude  and  the 
two  bases. 


CHAPTER  XV 

FRACTIONS 

71,  Algebraic  fractions.    The  expression  --?  in  which  a 

and  h  represent  numbers  or  polynomials,  is  an  algebraic 
fraction.  It  is  read  "  a  divided  by  ^,"  or  "  a  over  5."  A 
fraction  is  an  indicated  quotient  in  which  the  dividend  is  the 
numerator  and  the  divisor  the  denominator.  The  numerator 
and  denominator  are  often  called  the  term%  of  a  fraction. 

Certain  operations  upon  fractions,  such  as  multiplying 
both  numerator  and  denominator  by  a  number  (raising  to 
higher  terms),  and  dividing  both  numerator  and  denomi- 
nator by  a  number  (reducing  to  lower  terms),  are  often 
necessary  before  the  processes  of  addition  or  subtraction  of 
two  or  more  fractions  can  be  performed. 

The  change  of  a  fraction  to  lower  or  to  higher  terms, 
and  the  addition  and  the  subtraction  of  fractions  in  both 
arithmetic  and  algebra,  depend  on  the 

Principle,  The  numerator  and  the  denominator  of  a  fraction 
may  he  multiplied  hy  the  same  expression  or  divided  by  the 
same  expression  without  changing  the  value  of  the  fraction, 

3      3.4      12         ,  18      18^6      3 
^^^^  4  =  471^16'  "^^30^30T^=r 

_,..,,  a      a '  n      an  ,   «      a  -^  n      a/n 

Similarly,      -  = =  -—  ?  and  -  = =  ——  • 

b       b-n       bn  b       b  -^  n       b/n 

!  4    6  n 

Since  -  >  -  >  and  -  are  each  equal  to  1,  each  of  the  four  preceding 
4    6  n 

illustrations  is  really  a  multiplication  or  a  division  of  a  fraction 
by  1.  This  produces  no  change  in  the  numerical  value  of  any  fraction, 
though  it  may  change  its  form. 

148 


REACTIONS  149 

ORAL  EXERCISES 

Eead  the  result  of  multiplying  both  numerator  and  denomi- 
nator of  each  fraction  by  the  factor  on  its  right : 

2.         3.  -•      5.         5.  -•      4.         7. 


3  *  7  *  X  '  a  -{-  X 


3. 

7' 
a 

5. 

5. 

X 

a 

4. 

4. 

z 

3. 

6. 

T 

X. 

2.  -•      3.         4.  -•      3.         6.  -'      X.         8. 

o  z  0  a  -[-  X 

How  are  the  fractions  on  the  right  obtained  from  those 
which  precede  ? 

n    1    3  ,.2     8  ,^    4     8a 

2    6  5    20  9    18  a 

in    1     ^  .«    3     ^  .A       ^       21a^ 

5    15  7    14  10  aj    30  cc^ 

Eead  the  result  of  dividing  both  numerator  and  denominator 
of  each  fraction  by  the  number  on  its  right : 

2.   21.    2^ 


?, 

5 

^n 

15. 

4' 
3 

2. 

17. 

20' 

5. 

19. 

"6 

a 

lf>. 

3. 

18. 

6 

?,0. 

9 

18 

ab 

X  -\-  xy 


a.        22.  — ' — r-     a. 

a  —  ab 

How  are  the  fractions  on  the  right  obtained  from  those 
which  precede  ? 


'^'-  w  I 

or     24     4 
'^^•30'5- 

27.  "f ,  ^ 

0.    12    2 
^*-  18'  3- 

26-  7'?- 
a<^    b 

28     ^"^^ 
6icV 

72.  Reduction  of  fractions  to  lowest  terms.  A  fraction 
is  in  its  lowest  terms  when  no  factor  except  1  is  common 
to  both  numerator  and  denominator. 

Cancellation  is  the  process  of  dividing  the  numerator  and 
the  denominator  of  a  fraction  by  a  factor  common  to  both. 


150  FIEST  COUESE  IN  ALGEBEA 

EXAMPLES 

Reduce  to  lowest  terms : 
54a«Z>V 


1. 


144aZ.V  3     ^,^3 


54  a%^c''  _X-j^-//^/_  3  a%^ 
^''^''^'°''*    144aZ>M  ~  ^.  ^  .///^Z -  T7^~  * 

4a«-324a^ 


6  a^- 36  6^3-162  a  2a3 

Solution  4  a«  -  324  a^        _'^g^(a^  +  9),faH-S7,(a-^-^ 

3 

_2a3(a2^9>^ 

3  (a2  +  3) 

The  pupil  should  note  that  a  factor  which  occurs  one 
or  more  times  in  both  numerator  and  denominator  of  a 
fraction  can  be  canceled  only  the  same  number  of  times. 

For  reducing  a  fraction  to  its  lowest  terms  we  have  the 

IRule.  Separate  the  numerator  and  the  denominator  into 
their  prime  factors  and  cancel  the  factors  common  to  both. 

Cancellation  as  used  in  the  rule  means  an  actual  divi- 
sion of  the  numerator  and  the  denominator  by  the  same 
expression.  Therefore  only  factors  which  are  common  to  the 
numerator  and  the  denominator  can  he  canceled. 

The  terms  (the  parts  connected  by  plus  or  minus  signs) 
in  polynomial  numerators  and  denominators,  even  if  alike, 
can  never  be  canceled.    For  example,  it  would  be  incorrect 

5  +  2 

to  "  cancel  "  thus :  ~ ^  ->  as  the  resulting^  fraction  would 

6  +  ^  ,  . 

be  I  instead  of  the  true  value,  |^.    Similarly,  in  the  fraction 

o^    I    /Y  _L,  A  0^ 

— -  no  cancellation  is  possible. 

«/  +  a  +  8  .^  ^ 


FEACTIONS  151 

We  have  seen  that  we  may  multiply  or  divide  both  numerator 
and  denominator  of  a  fraction  by  the  same  number  without  affecting 
the  value  of  the  fraction.  But  we  should  never  forget  that  adding 
the  same  nwnher  to  or  subtracting  the  same  number  from  both  niimei^ator 
and  denominator  changes  the  value  of  the  fractiori.  Also,  squaring  both 
numerator  and  denominator  leads  to  a  different  value.  Compare  this 
statement  with  the  operations  that  may  be  performed  on  each 
member  of  an  equation  as  given  on  pages  39-40. 

EXERCISES 

Reduce  to  lowest  terms  : 

,     36c^  ^    21cd''e 

54  a^ 

^      52a^2 


130  cc^ 

84  m^ 

126  mn 


xf 


5. 

Xlf 

2xhj 

6. 

5  alP-G 

Zac^ 

rr 

lOa^r^ 

45  cV 

108  mV 

252  md'' 

2a 

a'-\-ah 

Ac'-Scd 

12  c 

l^xhf-  ^xy^ 

15xif 

da'-ea 

3a-2 

4.x'-  25 

10. 

11. 

12. 
13. 


"   25  ax^  ^^*  4x2-20::c  +  25' 

x^  —  A9x 
22     .  OQ 

'  4.2x^-27x^-h3x'^  3a^-a^  +  6a-2 

2a^y  -\-2  a^xy  +  2  acy  +  2  cxy 
2ax  -{-2o?  —  ay  —  xy 
^^     w?  —  n^  ^2  _  4  ^4  _  ^4 

25.   -1 V  27.  •; 7^.'  29. 


15. 

9_6:^4-^' 

9-^2 

16. 

9a;2-l 

9^2  _  9^  ^2 

17. 

10a2-2a 

15a2-|_7a-2 

18. 

c^-4c 
c^-8c2 

19. 

c2_6c  +  8 

c^  +  c  —  6 

20. 

2a^3_^;r2-3aj 

3aj*-3a;2 

21. 

50x-2a3« 

a;«  +  8cc2H-15a; 

a^- 

^2  _^  2  a  -  2 

,,,3_^3  -••  («_2)2  '^"^  a^  +  3aV4-2a;* 


^8_2^2  --       ^2_J_  --•       9^2  __  ^2 


RE 


152  rmsT  couESE  m  algebra 

73.  Lowest  common  multiple.  The  lowest  common  mul- 
tiple (L.C.M.)  of  two  or  more  arithmetical  or  algebraic 
expressions  is  the  expression  having  the  least  number 
of  factors  which  will  exactly  contain  each  of  the  given 
expressions. 

If  twQ  or  more  polynomials  have  no  common  factor 
other  than  1,  they  are  said  to  be  prime  to  each  other. 

Thus  5  c?b  and  4  o^y  are  prime  to  each  other,  as  also  are 
2  rc^  +  4  a;  and  o^  —  ^,  On  the  other  hand,  ar^  —  9  and 
jc^  —  6  rK  +  9  are  not  prime  to  each  other,  since  they  contain 
the  comnfon  factor  :z;  —  3. 

ORAL  EXERCISES 

Without  separating  the  expressions  into  their  prime  factors, 
find  the  L.C.M.  of  the  following : 


1. 

4,6. 

6. 

8,12. 

11. 

5,  10,  15. 

16. 

8,  10,  20. 

2. 

5,10. 

7. 

7,14. 

12. 

6,  10,  12. 

17. 

8,  20,  40. 

3. 

6,8. 

8. 

10,  15. 

13. 

6,  8,  12. 

18. 

3,  12,  15 

4. 

6,9. 

9. 

12,  18. 

14. 

7,  14,  28. 

19. 

4,  8,  16. 

5. 

8,  10. 

10. 

4,  6,  8. 

15. 

7,14,21. 

20. 

6,  18,  24. 

EXAMPLE 
Find  the  L.C.M.  of  36  a%\  72  a%\  and  108  a%^c\ 

Solution.  36  a%^  =  22  .  32  .  a%^, 

72  a%^  =  23  .  32  .  a%\ 
108  a%^c^  =  22  .  33 .  o^y^cK 

Since  the  L.C.M.  must  contain  each  of  the  expressions,  it  must 
have  2^  as  a  factor.  It  will  then  contain  22,  which  occurs  in  both  the 
first  and  the  third  monomials.  Similarly,  the  L.C.M.  must  contain 
as  factors  3  3,  a^,  &^,  and  c^. 

Therefore  the  L.C.M.  is  2^  •  '^^a%^c^,  which  equals  216  a%^c^. 


FRACTIONS  153 

The  method  of  finding  the  L.C.M.  of  two  or  more 
expressions  is  stated  in  the  following 

Rule.  Separate  each  expression  into  its  prime  factors. 
Then  find  the  product  of  all  the  different  prime  factors, 
using  each  factor  the  greatest  fiumber  of  times  it  occurs  in 
any  one  expression, 

EXERCISES 
Find  the  L.C.M.  of  the  following  : 

1.  96,  36,  40.  11.  %a%,  ^  ah\  30  &V. 

2.  45,  105,  175.  12.  A.c%  10  df,  24.cdh\ 

3.  50,  44,  110,  275.  13.  2  a\  ISax'^F,  A.^x^'k. 

4.  a^Xy  ax^,  aV.  14.  x^  —  xy,  2x^y^, 

5.  mn,  m^,  mV.  15.  a'^  -\-  ah,  2ah. 

6.  2x,  6x%  4.x\  16.  5c -10,  10c. 

7.  3  r,  12,  15  A  17.  b'' -  hk,  Pk, 

8.  4  a,  6  a^  8al  18.  ac^,  3a% 

9.  2  ax%  3  a\  a^cx^  Qa^c-d  ac\ 
10.  4tx%  ^x^yz\  10  xz".                   19.  a"  +  ah,  ah  +  h\ 

20.  3x«  +  6xV  +  3ccy2^9aJV-9^y^  6 cc^  - 12 ccf?/ +  6 a^y. 
Solution.    3  :r3  +  6  :?;2y  +  3  a;?/2  =  3  a:  (a:  +  y^ 

9  x^y  —  9  xy^  =  3^  xy  (a;  +  ?/)  (a:  —  y) 
6  x*  -  12  :r3y  +  6  xY  =  2-dx^(x-  y)^. 
Hence  the  L.C.M.  is  2  •  n^x^y(x  +  yfix  -  y)\ 

21.  a'-ah,2a-2h,2a'-2  ah. 

22.  ax  —  ay,  x^  +  ^y?  ^  —  y^- 

23.  a^  -  4,  a^  -f-  3  a  +  2,  a^  -  a  -  2. 

24.  4.x^-9,  2x^  +  7x  +  6,  2x^-\-x-6, 

25.  x^  -  1,  cc^  -  1,  cc^  +  2cc  +  1. 

26.  6^2  _  9^  2  a^  +  6  a^  4- 18  a,  2  a^  -  54  a. 

27.  x^-^x,  4.x'-\-2x,2x'-\-4.x,  2x^-Zx-2, 


154  FIRST  COUESE  IN  ALGEBRA 

74.  Equivalent  fractions.  Two  fractions  are  equivalent 
when  one  can  be  obtained  from  the  other  either  by  multi- 
plying or  by  dividing  both  numerator  and  denominator  by 
the  same  expression. 

For  example,  -  and  -—  are  equivalent  fractions :  also  —  and  - . 
5  10  x^  X 

The  lowest  common  denominator  (L.C.D.)  of  two  or  more 
fractions  is  the  L.C.M.  of  their  denominators. 


EXAMPLES 

Reduce  to  respectively  equivalent  fractions  having  the  low- 
est common  denominator  : 

i.  -g-,  g,  ana  20' 

Solution.  The  L.C.M.  of  the  denominators  is  60.  Multiplying  the 
numerator  and  the  denominator  of  the  first  fraction  by  12,  of  the 
second  by  10,  and  of  the  third  by  3,  we  obtain  Jf,  J^,  and  f  J 
respectively. 

3  a?  -y     ^y 

2-  o — ^  and   ~-' 

Solution.  The  L.C.M.  of  the  denominators  is  2^x'^yz^.  Multi- 
plying both  numerator  and  denominator  of  the  first  fraction  by  the 
factor  3  x^,  which  is  a  factor  of  the  L.C.M.,  but  not  of  the  denom- 

9  x^ 
inator  of  the  fraction,  gives  — — - — -•     Multiplying  both  numerator 

24  x^yz^  0 

and  denominator  of  the  second  fraction  by  the  factor  4?/,^,  which 

is  a  factor  of  the  L.C.M.,  but  not  of  the  denominator  of  this  fraction, 

20  y^z  9  x^  20  ?/% 

gives  zr—^ — -  •    Hence  the  required  fractions  are and 

24:x''yz^  ^  24:xhjz^  24:  x^yz^ 

^      ^a         ^      Sb 


Solution.    The  L.C.D.  is  18  a6V 

^,        4  a        4  a  .  2  ac  S  a^c  .   8h         8  ^  •  8  />2  24  b^ 

Inen— — -  =  — — — - —  = -— — — — ,and- 


9 /A'      9b^C'2ac      18  a&V  Qac^      Qac^-Sb^      IS  ab^c^ 


FRACTIONS  155 

Therefore,  to  change  two  or  more  fractions  (in  their 
lowest  terms)  to  respectively  equivalent  fractions  having 
the  L.C.D.,  we  have  the 

Rule,  If  the  prime  factors  of  the  denominators  are  not 
apparent^  rewrite  the  fractions  tvith  their  denominators  in 
factored  form. 

Find  the  L.O,M,    of  the   denominators  of  the  fractions. 
Multiply  the  numerator  and  the  denominator  of  each  frac- 
tion hy  those  factors  of  this  L,  CM,  which  are  not  found  in 
the  denominator  of  that  fraction, 

ORAL  EXERCISES 

State  the  lowest  common  denominator  of  the  fractions  in 
each  of  the  following  exercises.  Then  change  the  fractions  to 
respectively  equivalent  fractions  having  this  L.C.D. 


1. 

h 

\- 

6. 

h  tV 

11. 

h  h  i- 

16. 

h 

h  -ir 

2. 

1 

3' 

1 
4- 

7. 

TO'  T-g-- 

12. 

h  h  h 

17. 

h 

if- 

3. 

1 

3' 

1 
6* 

8. 

1        1 
T2'  T^- 

13. 

111 
4'    8'    12- 

18. 

h 

hh 

4. 

1 

4' 

1 

8- 

9. 

1        1 
15'    2  0- 

14. 

111 
TO'  T"§"'   2  0* 

19. 

2 
3' 

h  tV 

5. 

h 

tV 

10. 

2^'    3^- 

15. 

h  h  1- 

20. 

4 
5' 

9         4 
TO'  TP 

EXERCISES 

Change  the  following  fractions  to  respectively  equivalent 
fractions  having  the  lowest  common  denominator : 

X    y    z  5  a    Sax    2  ax^  3      2  c     5  a 

^'  3'4'5*  ^*  T'ir'"2r*        ^'  5^'Sd'lOd' 

^*    8  '  12  '   24  4    '    10  '     24    *  Ai/  6?/'  lOy' 

2a-\-h      a      5  a^  1  a      5e       4c 

3  61     '  2  a^ '  4  a  *  cde^    cd^e    c^d^e^ 

c-\-  d    So  —  d     4: a             ^^      1  x       4?/        2z 
8.   -T7^ — >  •  ^  ^   o    '  7rr~^'  10. 


12c        15c^      24c^  Sxy''    5xz    lOyz^ 


166  FIEST  COUESE  IN  ALGEBEA 

11        ^  * 

11.    -y  — u 

X  —  O    X  -\-  o 

Solution.    The  L.C.D.  is  (x  —  6)(x  +  5).  Multiplying  both  numer- 
ator and  denominator  of  the  first  fraction  by  a:  +  5,  and  those  of  the 

second  fraction  by  a;  —  5,  gives  ^^ ^ and  ^^ ^ , 

3. +  15        ,L-20  (y^i^-^^)  (.4-5)(.-5) 

12.    -y  -t:-  14. 


a-2    a  +  2  2x-5    2x-\-5 

a  h  ^  ^     —  2  c       S  c 

13.  7'  — —7-  15 


16. 


a  —  b    a  -j-  b  c  —  3c  +  3 

2a  a        2 


4:  —  a    4:  -{-  a    a 

Solution.   The  L.C.D.  is  (4  —  a)  (4  +  a)  a.  The  respectively  equiva- 
lent fractions  are 

2a(4:  +  a)a              a(4:-a)a  ^^^^  2 (4  -  a)  (4  +  a) 

(4  -  a)  (4  +  a)  a '  (4  +  a)  (4  -  a)  a  '  ^^^     a(4  -  a)  (4  +  a)  ' 

ic          2cc      2                   _  5c       -2c    3 

17.^ ; ,  ?-•  19.    -,  -,-•     , 

X  -\-  y    X  —  y    X  G  —  o    c  —  o    c 

2  a          3        —a             ^  m         — 2m2 

a  —  3    a  -\-  6       2  rn  —  n    m  —  n    m 

21.    -y 


a-2    Sa-6 

Hint.  Eewrite  the  fractions  with  the  second  denominator  in  factored 
form. 

a  5  ^2a5  +  33-2cc       x 


22.  -^ , 26. 

X  —  xy    X  —  y 

Sa  2a 

a^—P    a  +  b 

5  a       —4:  a 
24.  —„ — 77,^  7-  2«. 


9_^2     9_^3^    3_^ 
—  2x        3  5^ 


X  —  2    cc^— 4    x^—  5x  -{-  6 
2a-7  S-5a  2 


a^—9a    a^—5a^-\-6a    a  — 3 
a  2  a 


^^-  x^-y^'  x-y'  ^^*  a^-%'  a^ -\- 2  a" -\- 4.  a 


30. 
31. 
32. 


FEACTIONS  157 

5x  3x 


2^2+ 3;r    2x^-x-6 
—  4x  5x 


S-x    3  +  8^;-3cc2 
—  1  5x  -}-  y 

Note.  The  problem  of  operating  with  fractions  presented  great 
difficulties  to  all  the  early  races.  The  Egyptians  and  the  Greeks, 
even  down  to  the  sixth  century  of  our  era,  always  reduced  their 
fractions  to  the  sum  of  several  fractions,  each  of  which  had  1  for 
a  numerator.  For  example,  f  would  be  expressed  as  J  +  J.  The 
Romans  usually  expressed  all  the  fractions  of  a  sum  in  terms  of 
fractions  with  the  common  denominator  12.  The  Babylonians  re- 
sorted to  a  similar  device,  but  used  60  for  the  denominator.  In 
some  way  they  all  attempted  to  evade  the  difficulty  of  considering 
changes  in  both  numerator  and  denominator.  The  Hindus  seem  to 
have  been  the  first  to  reduce  fractions  to  a  common  denominator, 
though  Euclid  (300  B.C.)  was  familiar  with  the  method  of  finding 
the  least  common  multiple  of  two  or  more  numbers. 

75.  Addition  and  subtraction  of  fractions.  If  two  or  more 
fractions  have  the  same  denominator,  their  sum  is  the  frac- 
tion obtained  by  adding  their  numerators  and  writing  the 
result  over  their  common  denominator. 

-p  1      24,7      13         ,n,3n,5w      9n 

For  example,  -4--.  +  ^  =  _-,and -  +  —-  +  — -  =  ---• 
5555  a       a         a         a 

If  two  fractions  have  the  same  denominators,  their  differ- 
ence is  the  fraction  obtained  by  subtracting  the  numerator 
of  the  subtrahend  from  the  numerator  of  the  minuend  and 
writing  the  result  over  their  common  denominator. 

For  example,        =  -  ,  and = - 


9      9'  d      d         d 


If  it  is  required  to  add  or  to  subtract  two  fractions  hav- 
ing unlike  denominators,  the  fractions  must  be  changed 


158  FIEST  COUESE  IN  ALGEBRA 

to  respectively  equivalent  fractions  having  a  common  de- 
nominator; then  their  sum  or  their  difference  is  obtained 
as  explained  above. 

For  example,  to  find  the  sum  of  5^  +  f  +  yV  ^^  reduce  the 
fractions  to  respectively  equivalent  fractions  having  the  common 
denominator  40,  by  multiplying  both  numerator  and  denominator 
of  f  by  10,  of  f  by  5,  and  of  j^^  by  4.  The  fractions  become  f g^, 
f  g,  and  If  respectively,  and  their  sum  is  |g. 

In  adding  algebraic  fractions  with  unlike  denominators, 

as  --  and  -^  we  proceed  in  a  similar  way. 

Multiply  both  terms  of  -  by  d,  and  of  -  by  ?/,    The  fractions 

xcl  Tiy  x(l  ~l~  fiy 

become  —  and  -^  respectively,  the  sum  of  which  is •    Simi- 

yd  yd  yd 

^     ,      X      n      xd      ny        ,  .  ,  .     xd  —  ny 

larly, = -,  which  equals • 

y      d      yd      yd  yd 

The  pupil  should  always  reduce  fractional  results  to  their 
lowest  terms. 

ORAL  EXERCISES 

Find  the  algebraic  sum  of : 

^'    3+3-  10. 16.    — 


Ji 


V      y  ^v       ^'u 


^^._.  17.^^'-^      ^-^^ 


2.    -/-   +  -r-, 

r>    '    5 

Q      <Sill  AA«; 7*  A'*    — 7^ TT 

^'    9+"9'-                    hi)                             2r  2} 

•f.                     m       2m                     2x-jf  2y^x 

n         n                           6x  ox 

^   ^  ^*  ^^-  2x      2x  ^^-    x-\-y  x-Vy 

7      12     I     1_0                                                                                     -^  ^ 

8.  y-  —  \,             '   y^      y^            '    -^  —  //  x^  —  ?f 


a       a  '  4:X^      4  x^  '  ci?  ^y^      a^  ^  1? 


FEACTIONS  159 

EXABIPLE 

^.      ,.,     5c    ,   5a-2e      2c-4 

Simplity  7—-:,  H — 

^      -^  2a^  baa  3a 

Solution.    The  L.C.D.  is  6  a^c. 

PC      5a- 2c  _  2c -4  ^  5  c  ♦  3  c       (5  a  -  2  c)  a  _  (2c-4)2ac 
2  a^  6  ac  3  a  2  a^  •  3  c  Q  ac  -  a  3  a  •  2  ac 

_  15  c^  +  (5  q2  -  2  ac)  -  (4  ac^  -  8  ac) 

6  a'^c 
_  15  6'^  +  5  g^  -  2  ac  -  4  flc2  +  8  ac 

6  a^c 
_5q2  +  6  ac  -  4  ac'^  + 15  6'^ 
6  a^c 

Check.    Setting  the  original  expression  equal  to  the  final  result 
and  substituting  2  for  a  and  3  for  c,  we  obtain 


5c5a-2c      2c-4_5a2  4-6 

ac  -  4  ac2  +  15  c^ 

2a2          Qac              3  a 

G  a\^ 

15       1       1  _  20  + 
8       9      3  ~ 

30 

-  72  +  135 

72 

135       8       24       119 

72       72      72  ~  72  ' 

119       119 

72        72 

Therefore,  to  find  the  algebraic  sum  of  two  or  more 
fractions  (in  their  lowest  terms)  we  have  the 

Rule,  Reduce  the  fractions  to  respectively  equivalent  frac- 
tions having  the  lowest  common  denominator.  Write  in  succes- 
sion over  the  lowest  common  denominator  the  numerators  of 
the  equivalent  fractions^  inclosing  each  numerator  in  a  paren- 
thesis preceded  hy  the  sign  of  the  corresponding  fraction. 

Rewrite  the  fraction  just  obtained^  removing  the  parentheses 
in  the  numerator. 

Then  combine  like  terms  in  the  numerator  and,  if  necessary, 
reduce  the  resulting  fraction  to  its  lowest  terms. 


160  FIEST  COUESE  IN  ALGEBEA 

EXERCISES 

Find  the  algebraic  sum  of : 


1. 

f  +  f. 

8. 

a 

+  i|-                 15-  *l 

+  1-3^. 

2. 

1^0 

+  tV 

9. 

M 

-u-       16.  e 

-M  +  H- 

3. 

A 

-iV 

10. 

If 

+  2||.              17.  if 

11          25 
—  -39  —  5  2- 

4. 

T^^ 

-tV 

11. 

M 

-M-            18-11 

1  9     1     27 
~  4  5  +  5  0- 

5. 

A 

+A- 

12. 

11 

+  if            19-  if 

-H  +  ih 

6. 

5 
6   " 

-tV 

13. 

r 

-IA  +  3J2.      20.  li 

+  2-H. 

7. 

2  V 

-M- 

14. 

ii 

-il  +  2iV   21-  M 

-M  +  i^ 

22. 

3a      4a. 

2   "*"    3 

• 

25    ^"^  1  ^^ 
^^-    7   +  21 

3x 
14 

23. 

2a      3a 
5   +   2 

11a 

■^  10  ■ 

„  +  2      3a -2 

2C.       g 

10 

24. 

7a       6a 
10       15 

3a 
"•"   5 

„„    5a. -3 

2aj-7 

"•       12 

14 

28. 

3a- 
4 

^ 

2  —  a  ,   7a 

6       '     8 

29. 

X  —  5 

a 

3a;  +  2a      a— 7a; 

10 

18                20 

30. 

4m  — 
.    6 

1_ 

a  —  m,     m  +  Sa  —  S 

8                30 

31. 

5+    2   . 

„„      7         2,3 

36.  5-^ +  -i- 

3  a''      ax      X 

32. 

2 
3« 

.+    5 
•       2a^ 

10  a 

4  a' 

2a        11         5a; 
3  a;      2aa;^        a 

• 

33. 

2a         3 

ba^      10  a 

5 
60" 

,„    3a4-l       4a  +  3      5  — a 

^°-       5a              3«^ 

6  a 

34. 

ax        2 
X       a       ax 

^^    4a;2-5       2-3 
'^-       3a;^              2  a; 

x       3x-l 

35. 

a 
c 

1        3 

ac      2  a 

2a;        3?/-l 

4cc2/  +  3 

*      5a;V        10a;2/'' 

isa^y 

41 


FRACTIOKS  161 

3a  +  2  26^-1 


^^2  „  4        a^_Sa-10 
Solution. 
3a  +  2  2a-l  3a  +  2  2a-l 


a2-4       a2-3a-10      (a  +  2)  (a  -  2)       (a  +  2)  (a  -  5) 

_       (aa  +  2)(a-5) (2  a  -  1)  (a  -  2) 

~  (a +  2)  (a -2)  (a -5)       (a  +  2)  (a  -  5)(a  -  2) 
^  (3  a  +  2)  (g  -  5)  -  (2  g  -  1)  (g  -  2) 

(a  +  2)(a-2)(a-5) 
__  3  q2  - 13  a  -  10  -  (2  c?2  -  5  g  +  2) 
""  (g  +  2)(g-2)(g-5) 

^3g2-13g-10-2g2  +  5g-2 
(g  +  2)  (g  -  2)  (g  -  5) 
g2  -  8  g  -  12  g^  -  8  g  -  12 


(g  +  2)  (g  -  2)  (g  -  5)  g^  -  5  g^  -  4  g  +  20 

Unless  other  directions  are  given,  the  denominators  should  be 
retained  in  factored  form  throughout  the  process. 

Check.   Let  g  =  3. 

3g  +  2  2g-l  g2-8g-12 


g2-4       g2-3g-10      (g  +  2)  (g  -  2)  (g - 

-5) 

9  +  2           6-1           9-24-12 

9-4      9-9-10      5-1.  (-2) 

11         5         -27        27      27 
5       - 10      - 10  ^^  10      10 

In  checking  work  in  fractions  such  values  must  he  chosen  for  the 
letters  as  loill  make  no  denominator  zero.  This  prevents  the  substitution 
of  —  2,  2,  or  5  for  a  in  the  foregoing  example. 

2y  +  7  2y'-35 


42. 
43. 

44. 

12 

x^-9 

7 

6^2-49 
5 

4.x'' -1 
4c4-3 

x^-5x  +  6 

2 

a^-6a-7 

3x 

8a;'- 1 

1      ^' 

46. 


3  +  2/       f-lly-42 


c  +  c 


_    5a+l        7.2 

47 \- —  • 

a-\-3        2a^3 

48.  -^ ;= 1 -' 

x^  —  7  X      X      X  —  7 

49.1^  +  ^-4- 


162  -FIRST  COUESE  IN  ALGEBRA 

5c^         3c +  2  5  c 


51. 


4^2-1      26'  +  l      2c-l 

m^+  m  3  ??i  -f-  7  m  —  3 

m^  —  8       m^  +  2  7?2,''^  +  4  ??2-      ??i  —  2 


_    a2+2aZ^  +  />2  aZ>  2^ 

52.  1^ 7^ +  ■ 


c^^  —  b^  d^  +  a^ .      ab  —  Z/^ 


■ic  —  2      X  —  2      x^  -\-  X 
KA  ^         I  b  +  a  2b 

54.  T^ — 17  +  t;^ — o...  .  z.  + 


55. 
56. 
57. 


ab       a^—2ab-j-  h^       —  ab  -\-  a? 
cx^  2  ex         cx'^(c  —  2) 

c^  —  x^      c^  —ex         c^  —  cx^ 
a  +  c       a^  —  ac  '\-  c^        4  a^c 
a  —  c      a^-\-  ac  +  c^      a^  —  c^ 
x''-{-Sx-\-9      x-3  54 


x''-3x  +  9      x-i-3      x^-\-2T 
-5x  -\-  x^  2x 


^^'  2x  +  5'^  2x^-{'15x  +  25       x  +  5 

76.  Changes  of  sign  in  a  fraction.  The  sign  of  a  fraction 
is  the  plus  or  minus  sign  placed  before  the  line  separating 
the  numerator  from  the  denominator.  Hence  there  are  in  a 
fraction  three  signs  to  consider:  the  sign  of  the  fraction, 
the  sign  of  the  numerator,  and  the  sign  of  the  denominator. 

Now  in  division  the  quotient  of  two  expressions  having 
like  signs  is  positive,  and  the  quotient  of  two  expressions 
having  unlike  signs  is  negative. 

Therefore     +±|  =  +  4;  +^  =  +4; 

Or,  m  general  terms,   H =  +  — -  = = 

+  6  —  h  +6  — 6 


FRACTIONS  163 

These  examples  illustrate  the 

Principle.  Without  altering  the  value  of  a  fraction  the  fol- 
lowing changes  in  sign  may  he  made: 

(a)  The  sign  of  the  numerator  and  the  sign  of  the  de- 
nominator, 

(h^  The  sign  of  the  numerator  and  the  sign  before  the 
fraction, 

(c)  The  sign  of  the  denominator  and  the  sign  before  the 
fraction. 

Hence  any  fraction   may  be   written   in   at   least  four 

ways,  if  proper  changes  of  sign  are  made. 

^,  Zx        —  ?>  X  — Sir  3a; 

Thus         H =  --^ =  —   z  =  —  -z 

x—b      o—x  x—o  b—x 

Similarly, 
a -2b  .      2b -a  2b- a  a- 2b 


■b-\-2tc  b~a—Zc  a  — 6  +  3<?  b  —  a—'6c 

The  pupil  should  note  particularly  that  changing  the  sign  of  the 
numerator  involves  a  change  of  sign  in  each  term  of  the  numerator. 
Similarly,  a  change  of  sign  of  the  denominator  involves  a  change  of 
sign  in  each  term  of  the  denominator. 

EXERCISES 

Write  each  of  the  following  fractions  in  three  other  ways : 

2a-h 


a 

n 



8 

1. 

H. 

5. 

..- — 

-h 

d 

X  - 

-y 

—  G 

2 

X 

-2 

?!. 

4. 

— . 

6. 

2d 

a  — 

c 

2x 

-3 

c-Zd 
'    d^-c^' 
x 


'  5x  —  6  —  x^  '        jl  —  x" 

77.  Changing  signs  of  factors  of  denominators.    In  their 

a  —  3         2 
present   form  the  L.C.D.   of  ;=:  —  ;; is   apparently 

a  — 7      7—  a 

(a  — 7) (7— a).    But  inst^?t4  pf  taking  these  factors  it  is 


a 
a 

5 

3 

7' 

7 

2 
—  a 

= 

a 
a 

E 

3 

7 

+ 

2 

a  — 

7~ 

a 
a 

-1 

-7 

X 

^ 

+ 

4- 
3- 

-2x 

—  X 

= 

X 

X 

"3 

+ 

2x 

-4 

3x- 

-4 

X 

X  - 

-3 

a:  ~ 

3 

3 

5 

3 

+ 

_5_ 

-  etc. 

164  EIEST  COUESE  IN  ALGEBEA 

better  to  apply  the  principle  of  section   76  and  rewrite 
the  fractions  so  that  the  denominators  will  be  identical. 

Thus 
And 

Similarly,       ^^  __  ^      ^_^      (,  +  2)  (.  -  2)   '  x-2 

In  addition  and  subtraction  of  fractions  whenever  a 
factor  occurs  in  one  denominator  which  is  the  negative  of 
a  factor  in  another,  the  form  of  one  of  the  fractions  should 
be  changed  according  to  section  76.  This  will  greatly 
simplify  the  work  and  decrease  the  likelihood  of  errors. 

EXERCISES 

Simplify  the  following : 


i: 

7       .       3                            ^    x''-2a  ,   a-\-x^ 

• z  +  ^          •                       3.  -  ■          -  + 

X  —  5      5  —  X                              X  —  a         a  —  X 

2. 

a            ^b                            ^    x-2      2  +  x   1      ^x 

a  —  b      b  —  a                        ""3  —  icoj  —  3'x  —  3 

a          a^  -\-5a 
5 -a  ^~  a^-25 

a      .  a2  +  6  a        -  a    ,         a^  +  5  a           . 

Hint.              +                 — V etc. 

6  _  a      a2  -  26      a  -  6      (a  +  5)  (a  -  5) 

6. 

2         •    9                 2a^5      2a-5       46^^+1 
c2_9      3-c         ®'   5a+l    '    l-5a       25a^-l 

7. 

3                4                    a-  b             10 ab             a-\-b 
j^x      ir2-49      ^'   3a  +  2b      4:P-9a^      3a-2b 

2aj+l       -6a^+l       2a;-l 
^"'   2x-l        1-4:^2        2^+1 

_ll^  +  46-6a2      2a-5      T--4.a 
^^-        a^^5a-14.        '     a+7     '     2^a 

FEACTIONS  165 

78.  Reduction  of  a  mixed  expression  to  a  fraction.    The 

mixed  number  3|  really  means  3  -f  |.    It  is  equal  to  |^  4-  f 
or  J^  + 1^  =  J^-.    The  corresponding  case  in  algebra  is  rep- 
resented by  a-] — 
c 

,  b      a  ,  h      ac      h      ac  +  b 

Now  a  +  -  =  r  +  -  = h  -  = 

c      1       c       c       c  c 

Similarly, 

""^      "^a  +  2~     1  a  +  2  a  +  2  a  +  2  a  +  2       * 

The  process  involved  in  this  operation  is  really  nothing 
more  than  the  addition  of  two  fractions,  one  of  which  has  the 
denominator  1.    Hence  the  method  of  section  75  applies. 


ORAL  EXERCISES 

Reduce  to  fractional  form:- 


o2 


l-2  +  i  15.  1  +  |. 

2.  3  +  ^  *  4 


^-  15.  1  +  ^-  23.  y  +  s\ 

1  6  4 

':'/]■        .6.2-5.        «.(|)'- 

•  W        5' 


4.  5  +  ^ 


a 


6.  6  +  f.  17.  a-^-  25.  y^^         ^ 

6.  1  +  3.  ^^Y      ^ 

7.  i  +  3.  18.  c---  26. 

8- ¥-4. 


7^ 


27.  2  + 


9.  J53._3  19.  r--.  r  +  1 

10.  2-|.  20.  .^-i.  28.5-     ^ 

11.  7-f 


1-  s  -  1 

,2  . 


12.5  +  1  21-^-r  29-  -  +  ^- 

13.  7-|.  /,.V  2 

14.12+1.        '^^•^-y-      ^'-y-j^^- 


9. 

^- 

(I)- 

10. 

x"- 

-(W 

11. 

2*2 

2  a'' 
3 

166  FIRST  COUESE  IN  ALGEBRA 

EXERCISES 

Write  the  following  mixed  expressions  in  fractional  form: 
X  n 

3.  2c  +  ^.  ■  7.  .+|. 

4.  5a; -3-  fi.x^-^-  12.  «  +  2  +  2^. 

d  5  2 

13.  c  — c?4 -•  17.  m^  +  mre  +  TO^ 

c  +  a  in  —  n 

14.  0^-22/-^^^.        18.  x'-xij-'^^^  +  zf. 

15.  a2  +  32  — ^ — i7— ^-        19.  5  — 7^3— -z ^ — 2-x, 

2  25  -\-  5x  +  x^ 

16.  5  a  -  7  —  ^ — ;  •         20.  — ^  -  3  +  ; 


5a  +  7  a;-2  2cc-f-3 

r^  —  18 

21.  a^«  +  4a:-^^^ ±S.^Si-2x^ 

X  —  2 

Hint.    Removing  parentheses  and  combining,  we  get 

ft  J.  2 2  0  +  1  H =  d  —  0 etc. 

2b-  1  6  +  5  26-1        6  +  5 


FEACTIONS  167 

79.  Multiplication  of  fractions.  In  algebra  as  in  arithmetic 
the  product  of  two  or  more  fractions  is  the  product  of  their 
numeratois  divided  by  the  product  of  their  denominators. 


Thus 

i-f  =  ii 

Similarly, 

a    c  _  ac 
b"d~bd' 

and 

5.3  =  ^.^  =  i^. 

In  like  manner 

a  _  n    a      na 

If  a  factor  occurs  one  or  more  times  in  any  numerator 
and  in  any  denominator  of  the  indicated  product  of  two 
or  more  fractions,  it  should  be  canceled  the  same  number 
of  times  from  both,  thus  giving  the  product  of  the  several 
fractions  in  lower  terms. 

Thus  ^.^-  =  ^. 

p    c      c 

EXAMPLE 
Multiply  — by  j^^. 

Solution.        8ac^.l5^^£^.J^^5^. 

3  2ac^ 

To  find  the  product  of  two  or  more  fractions  or  mixed 
expressions  we  have  the 

Rule.  If  there  are  integral  or  mixed  expressions^  reduce 
them  to  fractional  form. 

Separate  each  numerator  and  each  denominator  into  its 
prime  factors. 

Cancel  the  factors  common  to  any  numerator  and  any 
denominator. 

Write  the  product  of  the  factors  remaining  in  the  numerator 
over  the  product  of  the  factors  remaining  in  the  denominator. 


168 


FIEST  COUESE  IN  ALGEBEA 


Simplify : 


ORAL  EXERCISES 


1. 

J- 

■h 

2. 

1 
'5 

1 

3. 

i 

■h 

4. 

f 

•1- 

5       5.12. 
0.     Q    "J  . 


6     -8_ 

D.     1  1 


22 


^      C      X 

9.  -  .  -• 


10. 


TT  •  TO* 
7-  (A)^. 


11.^.^. 


12.  -^.^• 


13. 


14. 


15. 


©' 


2x     10 


18. 


19. 


2x  -\-  c       xc^ 


x-2c 


Multiply ; 

1. 

2.  _ 

3      4.15      Q 


EXERCISES 


12 

_4 
21 


10 

12 
TO- 


10.  -V-  •  (6) 


4. 
5. 
6. 


TIT- 

n4       2  2      /9l\3 

10  V    24^^ 


12. 


13. 


14. 


15  ( 


15. 


16. 


4cc 
3x 


15  x' 
'  12  a' 
"*40x2 

18 
4.0  c^d' 

6a^ 


*^6   '  21- 


11 
11 


33 
T2- 


o4 


-m 


7     ol        1 
^-    ^T  •  2T 


If 


».  —  i^T  •  %r 

Q      14      ifi      32. 

y.  2T  •  ^^  •  TO 


17.  21ajV- 


14^ 


16  i 


9a^ 
hxH. 


19. 


20. 


21. 


'2x^ 

•  V5cV  *  \2d)  '    ac  ' 

\15  m/     \6  mW 


10  m 


(2x1     (Ml 
(6ay      125  ^>^ 


(10  Z»^)^     2(a«)2 


/BxV    6aP^    5^  /  8m"  V    /15 yi^ 

\2W  *10aj8'3ci**  VlOmTiV  *  V4m/ 


23. 


10  aj 


22. 

4 


\10  mTi"/ 
2^^-106^2 


a^-2a'-15'  2a^-Sa^'  a''-6a-{-9 


Solution. 


FEACTIONS  169 

a2-9  a -4  2a^-10a^ 


2- 2a -15    2a3-8a2     a^-Qa  +  9 


a-3 


Check.    Let  a  =  2. 

g^-Q a-4         2  g^- 10^2 

a2-  2  a  -  15  *  2  gs-  8  a2  '  a2  _  g  ^  ^  9 

-5      -2     -24_    1 
-15* -16*     1      ~-l' 
1    1 


3    8 


-  24  =  -  1. 
-1  =  -1. 


2x^  +  6       10 y^  a;" -2a; +  4    a;^  -  a;  - 

^^    c2-4      3a;-3^  ^„    cc2_3^_j_9     ^2_9 


28. 
29. 
30. 
31. 
32. 
33. 


a;-2/c2+6c  +  8  2a; -6        ic^+27 

aj2_7a;  +  10    2a;^  +  10a;2+50a; 


a;«  -  125 
;*  -  16      a; 


(x2-4)2    4  +  aj' 
c2-9  6^2_^4aa;  +  4a;2 


a^  —  4  a;^    ac-\-2cx  —  ?>a  —  ^x 
2a''-5ac-3c^      S  c -{- 9  a 

2a;2+10a;4-50    a;^-3a;-10    2- 


3a;^-12a;  4x^-500  2 

(a;^-4)(a;4-3)    a;^-a;-6    (a;  -  2) 
2a;2+12a;-|-18'    (a;^- 4)^    *    a;  +  3  ' 


4e^-12.a;  +  9a;^  a;  -  e 

(62+ ex +  x2)  (3  a;- 2*6)    '^        ^    262-56a;  +  3c 


170  FIEST  COUKSE  IN  ALGEBEA 

Hint.    (4h )( 1)  =  — etc. 

\        a2-l/\2a-l       /       a2_i     2a- 1 

The  student  should  note  particularly  the  difference  between  the 

procedure  in  multiplying   two   expressions   like   the   above   and   that 

followed  in  adding  them  or  in  subtracting  one  from  the  other.    (See 

Exercise  22,  p.  166.) 

36.  (-  -  4^  /?^±ii^\  f^^±lm±M 


.(3 


37.  {3x 


x^—  Sy^  /  \x^-f-  4^?/  +  4?/2 
168 


9x  +  15/\        3-\-x. 
38 

1\/.  3      \/l\       1 


a9.(2«-A)(6«^.^J(l)^. 


80.  Division    of   fractions.    In   arithmetic    f  ^  y  =  |  •  | 

—  21.    onrl     3_i_11  —  3.     1     —    3  A  kn     3  _:_  1  4  _  3  _i_  1  1  _ 

3   .     7—21 
1       11  —  $¥• 

^.     .,     ,       a      c      a    d      ad  ^  a  a    1       a 

Similarly,   --5--  =  -.-  =  — ;  and--j-w  =  -.-  =  — . 
0      d      0     c       be  0  b     n      on 

■    ,         a     /       n\     a      fed  -\-n\     a         d  ad 

Also     --^(c-\--]  = 


h      \        d)      b      \     d     /      b     cd-\-n      bed  -f  bn 

For  division  of  fractions  we  have  the 

Rule.  Reduee  all  integral  or  mixed  expressions  to  fractional 
form. 

Then  invert  the  divisor^  or  divisors,  and  proceed  as  in  mul- 
tiplication of  fractions. 


FRACTIONS  .  171 

EXERCISES 

Perform  the  indicated  operations ; 

1.  l-f.  1,—^-.  ^^    10^  .  15x^ 


2 


2x      X  14  2/2      21^* 


^    *    ^'  ^    2a       /-Sc\  ^^    12  ab^      lSa% 

A     1  2  _^  f_  ±\  ^    x^      2x  .  ^    4  a^x       2  ax 

15 3  2/  2/'  51c  17  C« 

21    •       ^-  _    2m       -m  _     /3a&V      18a»^» 

10. ^-  — 14. 


5. 


/3  ab\^ 


6.  Jf-5-(-if).  ^     '    3/1  \5c/  *  100c« 


V.2 


2^_^4^_^(3^  /  3  v.      256      ,     5a;« 

5a    *    15    *    12 a^'  \5xy  '  (lOx'^y  '  (SxY 

12cH  ^    8^2     IQde  2a^(2a^_^(7a^   /7V 

35  e^   *  7e^^'3c2c^«*  3^  *    12b^   *    15  6^ 
Hint.    See  Kule,  page  13. 


21. 

10-4a      5a-2a^ 

12^2      •         9^ 

22. 

(a-Sy       .  «'-9^ 

a2_4a  +  3   '   a^-a 

23. 

a^2_|.^_30           ^2-25 

x^  +  5x-6  '  x^-6x  +  5 

OA 

a^-1    ^  a'-^a^-^a'' 

a^-a  '  a^  +  3a-h2 

m*  -f  8  m 


25.   (2m2-4m  +  8)  -- ■ 


3m 


26.    U-fl-— -      3  4-T-^^ 7M-7; — -^• 


172  FIEST  COUESE  IN  ALGEBEA 

\3/       x)  '  \2x'-2xij  )  '  2(x-yf 

33. .    — ^— )-      3a-2^^+o-) • 

ax  \2b       3a/       L\  3a /    x   j 

81.  Complex  fractions.  A  complex  fraction  is  a  frac- 
tional expression  containing  one  or  more  fractions  either 
in  the  numerator,  or  in  the  denominator,  or  in  both. 


9x^—6xy-}-y^  \  ''  x  —  2y/      3x^-^Sxy  —  Sy^ 

(2mf      /    ,  16\  1     ,      4m 

m«  +  64    \^  "^mV   *  2m   *  4-m2' 

3aj-2    /^  ,    .    4\  9cc2-4 


EXAMPLE 


i  +  «-^ 


Simplify 


a      a' 


2 


a-3+- 


Solution.    Reducing  the  numerator  and  denominator  to  simple 

fractions, 

-       8_9^      g'-^  +  8  g  -  9 


^  ^  2       g^  -  3  g  +  2 

a  —  3  +  —      

a  a 

Performing  the  indicated  division,  the  right  member  becomes 

J/i.r^)(a  +  ^)  4  ^    g  +  9 

a 


FEACTIONS  173 


Check.    Letting  a  =  3, 


aa'a  +  9,                           393  +  9  ,. 
=  — '      becomes  = ,  or  4  =  4. 

2       a^-2a  ««29-6 

a-3+-  3-3+- 

a  o 

To  simplify  a  complex  fraction  we  have  the 

Rule.    Reduce  both  the  numerator  and  the  denominator  to 
simple  fractions^,  then  perform  the  indicated  division. 


Simplify : 


EXERCISES 


1-^  1-1-  ■    3-1  •       1  +  4 

y-(iy  (D-Cf)" 

■  3^  +  3  .  i  +  (i)^  •  (f )^  +  (f ) (f )  +  (f )^- 

6.  (1)^+1-2  ^     ar-ar    . 


•F 


-1  (ly+i-i+ay 


9,  ^  -I x-\ 

S"^!  12.-^.  15. 

1  +  ^ 

1  y 


0  —  ■ 
c 


V- 


"—^  13.  r-  16.:^-- 

o-t  f!  _  1' 

2  d''       (? 


'■'^'O' 


a^       Jy" 

h        a 

a 

X 

y     X  ^ 

z      ^ 

z 

2  +  5  +  ^ 

y                 X 

3  +  2.      ^y 
y       ^ 

174  FIEST  COUESE  IN  ALGEBEA 

—  (^  1 ^--l 

18. 24  ^       ^ 


\2 


1+2+2  1+1 


0^      2  x^      %f 

Bb^      2ah      5a^ 


1-^  i+A+^ 


19. 25. 

X  ^  2a^  b 

\2 


_  ^  _ 

bxy 


6  031/  a:  —  1       X  —  2 

20.  ^ 26. 

2x  Sx 


x-2      x-S 
2(b-{'d)-3(2a^c) 
2  ab  —  be  -\-  2  ad  —  cd 


/Sa-  bV 

21. 27. 

(^  -  3  ay  2 3_ 

4a^  2'a  —  c      b  -}-d 

x  —  2y      x-\-2y  /5a;  +  4yy_  5y 

22.  ^ 28. 

a;-2y      a;  +  2y  (5a;  +  4yf  _^^ 

x  2y  4:X  ^ 

lb  12- 6a 


a      a  '\-b  2  a^  —  a  —■  6 

23.  -i— •  29. 


^+i  2.  +  :     '' 


a      b  2a  +  3 


CHAPTER  XVI 

EQUATIONS  CONTAINING  FRACTIONS 

82.  Equations  containing  fractions  with  monomial  denomi- 
nators. If  fractions  are  involved  in  one  or  both  members 
of  an  equation,  it  is  necessary  to  find  a  number  or  a  literal 
expression  by  which  one  may  multiply  both  members  in 
order  to  get  rid  of  the  fractions.  (Compare  Problem  24, 
p.  145.)  This  process  involves  the  application  of  Axiom 
[II,  page  40,  which  is  the  only  principle  employed  in  this 
chapter  that  has  not  been  used  repeatedly  in  the  earlier 
work  with  equations. 

Especial  care  is  required  to  avoid  error  when  a  fraction 
which  has  two  or  more  terms  in  its  numerator  is  preceded 
by  a  minus  sign. 

ORAL  EXERCISES 

Solve  for  x,  stating  in  each  case  the  operations  employed : 

1.  1  =  1.      5.^  =  4.        9.  |-5  =  0.     13.^-8  =  0. 


X 


2aj       ^       ^ ^    'X      ^       ^      -.3ic 


2.  -  =  2.     6.  —  -  3.      10.  -  -  7  =  0.  14.  —  -f  3  =  0. 

3.7  =  1.      7.^  =  10.11.1-1-2  =  0.  15.^-10  =  0. 
4                      o                        o  6 

4.^  =  6.      8.5^  =  9.      12.  ^-M  =  0.  16.^4^  =  2. 
2                     4                       5  2 

176 


176 

riEST 

COURSE  IN  ALGEBRA 

17. 

'-^  =  3. 

21.  2%-^ 
5 

=  3. 

25. 

£-■ 

18. 

^  =  s. 

22.  1  =  -. 

26. 

£-■ 

19. 

2           ^■ 

23.3  =  1 

27. 

f.-». 

20. 

2a;-l       ^ 
3       =^- 

4 
24.  -  =  2. 

28. 

^-- 

29.  Using  the  least  multiples  possible,  free  Exercises  1-10, 
page  177,  of  fractions. 

EXAMPLES 

1.  Solve  the  equation  -  +  -  =  16. 

Solution.    Multiplying  both  members  by  15,  the  L.C.M.  of  the 
denominators,  and  canceling,  we  obtain 

5  X  +  3  X  =  240, 
or  Sx  =  240. 

a:  =  30. 

Check.    Substituting  30  for  x  in  the  given  equation, 
.3^  +  ^Y  =  16, 
or  10  +  6  =  16. 

r.  .    \.  \-       3,        ^,      5x-7      7 

2.  Solve  the  equation  -  (cc  + 1)  —  — — —  =  -  • 

Solution.    Removing  the  parenthesis, 

3x3      5a:-7^7 
4        4  6       ~3' 

Multiplying  both  members  by  12,  the  L.C.M.  of  the  denominators, 

and  canceling,  with  particular  attention  to  the  signs  in  the  third 

fraction,  we  obtain     ^         ^      ^  ^         -.  ^       r^o 
9  a;  +  9  -  10  a;  +  14  =  28. 

Solving,  X  =—  6. 


EQUATIONS  CONTAINING  FEACTIONS       177 
Check.    Substituting 


5  fo 

r  a:  in  the  given  equation, 

1)- 

-  25  -  7      7 
6             3 

-3  +  i^  =  I. 
3       3 

-9+16      7 

3             3 

7_7 
3      3* 

For  solving  equations  containing  fractions  with  mono- 
mial denominators,  we  have  the 

Rule.  Free  the  equation  of  any  parentheses  it  may  contain. 
Find  the  L,C,M,  of  the  denominators  of  the  fractions  and 

multiply  each  term  of  the  equation  hy  it,  using  cancellation 

wherever  possible. 

Transpose  and  solve  as  usual. 

EXERCISES 

Solve  for  x,  and  check  results  as  directed  by  the  teacher : 

a;  +  3      4.x  — 6       ^ 

a;-j-l   ,  x-2      ^+3_o 
9.  -2^  +  -3  4--^. 

.n    2         5         23 

^^-  r  + 4^  =  24' 

4         2 
11.  -x-\--x  =  2Q^. 
6  5 

^    x      X       X        ..  ^ft3.4         17 

^•5+3-15  =  ^^-  ^2-  2^  +  3'*'  =  T- 

4  9  XX 


1. 

2^3 

=  10. 

2. 

4"^8" 

=  3. 

3. 

X         X 

3~8~ 

=  5. 

4. 

.+1= 

=  6. 

X         X 


178  FIRST  COUKSE  IN  ALGEBRA 

15.5  +  ^  =  9.  18.^4-^  =  4. 

XX  Z                 6 

1 6    ?      1^  _  li  19    ^^+13  _  6-x  _ 

'  X      15      3x  3                4            * 

17.  —^--(x-3)=2.  20.  -(0.-1)+^^—==-. 

5.T+12              4     ,^  r^x           1            ^ 

21.— J -(2.  +  T)  +  -^0. 


=1  .  i/ocr:?\_^_:z22^ 


22.^ 


23.  -  +  x  =  a-{-l. 
a 

24.  -  -  :^  =  C. 

x_2^_^ 

25.  f^      —  ^     • 


26.  .;--^  =  lo,-3.)  +  -?;-- 

e       o  o 

3.T  +  4      2(2.r~l) 

28.  2  rr  +  2  ^^  + ^^^—  =  — : - 

a  (( 

29.  — ■ —  =  a'(lj  —  a). 

a  0 

2a -3x       5a-2x   ,   4.1       ^ 

30. + +-^  =  ^' 

b  a  b  a  o\) 


(x.-\-r>){x-(j)  =  x(u',-^^' 

34.  (4  0!  -f  3)  (6  +  x)  =  4  x  i^x  -  ^)  +  45f  • 


32 
33 


EQUATIONS  CONTAINING  FRACTIONS       179 


PROBLEMS 

1.  One  third  of  a  certain  iiuniber,  plus  -^2  ^^  ^^^^^^  nmnljer, 
equals  15.    Find  the  number. 

Hint.   !^  + -^  =  15. 
3      12 

2.  The  difference  between  |  of  a  (pertain  number  and  ^  of 
that  number  is  10.    Find  the  number. 

3.  The  sum  of  two  numbers  is  38.   One  tenth  of  the  greater 
number  equals  \  of  the  less.    Find  the  numbers. 

4.  The  width  of  a  rectangle  is  |-  of  its  length.    The  perim- 
eter is  200  meters.    Find  the  area  of  the  rectangle. 

(5.  AVhat  number  must  be  added  to  the  numerator  of  the/ 
fraction  -|  so  that  the  resulting  fraction  will  be  ^  of  the  number 
/  in  question  ? 

6.  One  half  of  a  certain  integer  is  ^  the  sum  of  the  next 
two  consecutive  integers.    Fiiid  the  first  integer. 

^7.  A  certain  even  integer  divided  by  12  is  equal  to  2V  ^^ 
the  sum  of  the  next  two  consecutive  even  integers.  Find  the 
first  integer. 

8.  What  number  added  to  both  ternis  of  the  fraction  ^ 
gives  a  fraction  whose  value  is  |  ? 

\9.  Separate  24  into  two  parts  such  that  1  of  their  differ- 
ence is  4. 

10.  One  fourth  the  difference  of  three  times  a  certain  num- 
ber and  4  equals  ^  the  difference  of  five  times  the  number  and 
4.    Find  the  number. 

11.  Separate  108  into  two  parts  such  that  their  quotient  is  |. 

12.  There  are  two  numbers  whose  sum  is  24.  If  their  differ- 
ence be  divided  by  their  sum,  the  quotient  will  be  -^  of  the 
greater  number.    Find  the  numbei-s. 


180  FIRST  COUESE  IN  ALGEBRA 

13.  A's  age  is  f  B's  age.  In  10  years  A's  age  will  be  twice 
B's  age.    Find  their  ages  now. 

14.  At  the  time  of  marriage  a  certain  woman's  age  was  -| 
that  of  her  husband.  Twenty  years  later  her  age  was  -|  of  his. 
Find  their  ages  at  the  time  of  marriage. 

15.  A  is  12  years  older  than  B.  Eight  years  ago  B  was  |^  as 
old  as  A.    Find  their  ages  now. 

16.  Jupiter  has  5  more  moons  than  Uranus,  and  Saturn  two 
more  than  twice  as  many  as  Uranus ;  Mars  has  7  fewer  than 
Jupiter,  and  Neptune  half  as  many  as  Mars.  These  planets 
together  have  26  moons.    How  many  has  each  ? 

17.  A  triangle  has  the  same  area  as  a  trapezoid.  The  alti- 
tude of  the  triangle  is  24  feet  and  its  base  is  12  feet.  The 
altitude  of  the  trapezoid  is  a  third  that  of  the  triangle,  and 
one  of  its  bases  equals  the  base  of  the  triangle.  Find  the  other 
base  of  the  trapezoid. 

In  solving  Problems  18  and  19,  the  student  should  make  a 
table  similar  to  those  shown  on  pages  100  and  101. 

18.  A  marksman  hears  the  bullet  strike  the  target  3  seconds 
after  the  report  of  his  rifle.  If  the  average  velocity  of  the 
bullet  is  1925  feet  per  second  and  the  velocity  of  sound  is 
1100  feet  per  second,  find  the  distance  to  the  target  and  the 
length  of  time  the  bullet  was  in  the  air. 

19.  A  gunner  using  one  of  the  best  modern  rifles  would  hear 
the  projectile  strike  a  target  2640  yards  distant  9|  seconds 
after  the  report  of  the  gun,  provided  the  projectile  maintained 
throughout  its  flight  the  same  velocity  it  had  on  leaving  the 
gun.    Find  this  velocity  if  sound  travels  1100  feet  per  second. 

83.  Equations  containing  fractions  with  polynomial  de- 
nominators. Although  no  new  principle  is  involved  in 
the  exercises  of  this  section,  they  serve  to  review  many  of 
the  most  important  proQ§§^§^  pf  ?ilgebra. 


EQUATIONS  CONTAINING  FKACTIONS       181 

ORAL  EXERCISES 

Clear  the  following  equations  of  fractions,  stating  in  each 
ease  the  operation  employed.   Do  not  solve  the  equations. 

^•^  =  ^- 

Solution.    Multiplying  each  member  of  the  equation  by  a;  -f- 1  we 
obtain  the  equation  x  =  3  a:  +  3. 

=  4. 

_5 

~2' 

=  1. 


2. 

X 

x-1 

3. 

2x 

1-x 

4. 

3x 

x^+i 

5. 

x-1 

x-2 

6. 

x  +  1 

x-5 

t 

x  +  1 

=  6. 


8. 

x-^7 
x-S 

5x 
3 

9. 

x-3 

x-{-2 

3 
4.x 

10. 

x  +  1 
x  +  2 

x-^2 
x-1 

11. 

x  +  2 
x-{-3~ 

x-2 
~  x-1 

19 

x-4= 

x-5 

x-{-3      x-\-2 


x 


3'       13.^-0^-1.  —  3^_i      2- 


14. 

1          2 

X        X  —  1 

15. 

X  —  1         X 

X          4 

16. 

X          5 

X  —  1         X 

17. 

x-2      1 
3x         X 

18. 

4:X                2x 

\-2x      1-^x 

1Q 

3  —  X          X 

EXERCISES 

Solve  the  following  equations  and  check : 

1.2  +  ^-^      11 


x  +  3       5 

Solution.  Multiply  both  members  of  the  equation  by  5  (a:  +  3), 
the  L.C.M.  of  the  denominators.  This  may  be  indicated  symbolically 
as  follows : 

By  means  of  this  form  of  expression  one  can  readily  see  what 
ought  to  be  canceled. 

Canceling,        10  (x  +  3)  +  5  (a;  -  1)  ==  11  {x  +  3), 
or  10a:  +  30  + 5a:-5  =  lla:  +  33. 

Transposing  and  collecting,  4  a:  =  8, 

whence  x  —  2. 


182  MEST  COUESE  IK  ALGEBEA 

2-1      11 


Check.  2  + 


2  +  3        5 
2  +  1  =z  1 J . 
.11  —  .1  1 


'*•  3  +  ;-t  -  ^-  ''•  a-  -  2  ~  2  *•  2.T  -  2  ~  "  - 

4  1  5x       3 -a; 

5. ^  =  -•  6.   ---  =  -— -  +  :<•. 

Solution.    The  L.C.M.  of  the  denoniiiiators  is  3(2  +  0^^).    Ileix^e 
from  an  inspection  of 


[t-I^^  "]»<=+"■) 


we  obtain  5  x  (2  +  x)  =  o  (8  —  x)  +  3  x  (2  +  x), 

or  10  :r  +  5  0:2  =:  9  -  3  2-  +  G  a:  +  3  ^,2^ 

Transposing  and  collecting, 

2x2  +  7a:-9  =  0. 
Factoring,        (2  :r  +  9)  (x  -  1)  =  0, 
whence  x  =—  %,  1. 

Check  as  usual. 
^    x-S        2x  ^,        2  0^-12   ,       X 

a;  r/'  —  Z  X  —  u        X  -f-  b        X  -f-  u 

8.    ;7  =  X.  15. 


a^-f3      ■• 

9. 

.T                      2  X 

5  —  .7*      4  X  —  5 

10. 

2              4 

x-j-1       4  +  a; 

11. 

1 

2  —  a^       3  —  X 

12. 

.r  —  0 

-; -7—  =  —  ()  i/*. 

16. 


X  —  4:         X  —  5 

4.r-l       7-2.r 


a;  +  2  :/•  -h  4 


4  3  X  -f  2 

17.  T, ;;  +  "    .^     =i^. 

2  i^  —  3  o 

X  +  G  _  5  +  2x 

^®-  3^: -7a. -5' 

3ir 4_        3a; -4 

~S   "  .7;  -  4  ~       8 


19. 


iQ    ^  +  7,3a;_  _2 1 l_-o 

^^*  '3-05^   2   ~^-       ^^'  4a;-+-l      5a;~2      3a;  +  4"'- 


EQUATIONS  CONTAINING  FRACTIONS       183 
47  7  3 


21. 
22. 


220      A{x-\-3)       5(x-^3) 
Ax  6  .r'^+ll 


a;  4"  3       2(ct'-f3)       3(a;-f-3) 


no       4--^     .     ^  ^ 

23.   r-f-r 


X'  —  5      5      5  —  X 
Hint.    Multiply  the  terms  of  the  last  fraction  by  —  1. 

24.  -A.  +  3  =  ^^—  25.  Jii^==|l=i^-.. 

a-  —  3  6  —  X  •  u;  —  J       J  —  x 

3.T            16              X       .      2x 
26. = 

a; -3      ^2-9      x -{- 3      9  -  ar 

Hint.  Rewriting  the  last  fraction  (see  Hint  after  Exercise  23),  and 
factoring  the  denominators, 

3x    _  16  _x 2x 

X  _  3  ""  (x  -  3)  (x  +  3")  "  X  -f  3  "  (x  -  3)  (x  4-  3)  * 

The  L.C.M.  of  the  denominators  is  (x  —  3)  (x  +  3).    Hence  from 

r  3x     _  16 x__  _  2x  1      _ 

[x  _  3       (X  -  3)  (X  +  3)  ~  X  +  3      (x  -  3)  (X  +  3)  J  ^^        )  (^  +    ) 

we  obtain  3  x  (x  +  3)  —  16  =  x  (x  —  3)  —  2  x,  • 

which  may  be  solved  and  checked  as  usual. 

In  solving  equations  containing  fractions  with  polynomial 
denominators  the  student  should  write  the  denominators  and 
their  L.C.M.  in  factored  form,  as  in  the  preceding  work.  With 
this  exception  the  rule  on  page  177  applies  to  all  equations 
containing  fractions, 

«».        1       ,       2  ^x 

27. h 


3-a;cr  +  3      x''-^ 

X  -{-1       x  —1       1—  x^ 

3  5a^-12  4 

29.  


x^-16       16 -x^ 


184  FIRST  COURSE  IN  ALGEBRA 

x^2      10-2:^2  7 


30. 


X  —  2  4:  —  x^  x"^  —  4: 


l-\-x       x  —  2,  X —1 

31.  ~  r  ■ 


X      2  —  X      x^  —  5x  -{-  6 
„^      0^-4     .  2;:c-15      8a;2-20ir-31 

OO       -L.  ^=  , 

2x-5       2aj  +  4        4a52-2(r-20 

84.  Equations  containing  decimals.  The  method  of  solv 
ing  an  equation  containing  decimals  is  illustrated  in  the 
following : 

EXAMPLE 
Solve  the  equation  .S  x  -{-  .7  =  4:.2  —  .05  x. 
Solution.   Multiplying  each  member  of  the  equation  by  100, 

30  x  +  70  =  420  -  5  X. 
Transposing  and  combining, 

35  X  =  350. 
a:-:  10. 

Check.  .3  X  10  +  .7  =  4.2  -  .05  x  10. 

3  +  .7  =  4.2  -  .5. 
3.7  =  3.7. 

In  equations  containing  fractions,  if  decimals  occur  in  any 
denominator,  multiply  both  numerator  and  denominator  of 
such  a  fraction  by  that  power  of  ten  which  will  reduce  the 
decimals  in  both  the  numerator  and  the  denominator  to 
integers.  Then  clear  of  decimals  and  proceed  as  in  the  fore- 
going example.     (See  Hint  after  Exercise  20,  p.  185.) 

EXERCISES 
Solve  and  check  the  following  equations : 

1.  Ax  =  6.  4.  .75  -  .7 a:  =  .26. 

2.  ,3x-\-.5  =  .8.  5.  .92  +  ,Sx  =  5.12. 

3.  .3  X  -f-  4  =  .25.  6.  3.75  =  2.15  -  .5  x. 


EQUATIONS  CONTAINING  FRACTIONS       185 

7.  .06ic- 4.5  =  1.68.  9.  .04aj  =  .Ice  +  2.4. 

8.  .03x  +  .16  =  .5S.  10.  .8aj-2.7  =  .55x. 

11.  .15x-Ax  =  Sx-A9.5. 

12.  1.7  :r  + 3.14  =-9.66 -1.5  a;. 

13.  3  X  +  7  -  1.25  aj  =  8.845  +  .52  x, 

14.  1.7  X  H-  .17  -  .03  a;  =  4.73  +  1.1  x. 

15.  .12  (2x-\-  .5)  -  .2 (1.5  a^  -  2)  =  .4. 

16.  6(3:zj-l.l)-8.4(.7a:-3)=6aj4-.24. 

'Jx_']^_'J^  .3  a; -6.2      6.75 -.4  a;      3.5 

4  3    "^    6   '  4  5  "■   2  * 

.14^^  +  3.2      a; -.75      2Ax-1.5 


19. 


6  5  10 


20.^^^  +  i^  =  3.9.   , 

Hint  .   Multiplying  numerator  and  denominator  of  each  fraction  by  100, 
600X-269      150x_ 

40  24    ~    *  * 

Solve  and  check  as  usual. 

To  avoid  the  possibility  of  repeating,  in  the  check,  a  numeri- 
cal error  made  in  the  solution,  the  check  should  be  performed 
by  finding  the  value  of  each  fraction  separately,  without  clear- 
ing of  fractions. 


21. 


1.8a; -2  ^5.7 -.7  a; 
1.7       ~        1.8 


^^    3.2  a;   ,   4.5  a;      ^  ^r,   ,   n 
22.  —77-  +  TTTT  =  1-52  -i-  6x. 


^^    .3(3 -Ax)      M(.2x-6)_^^ 
.16  .8 

^-       6.25  14        +-^^^- 


186  FIRST  COURSE  IN  ALGEBRA 

Note.  The  introduction  into  Europe  of  the  Arabic  notation  for 
nurnbers  was  one  of  the  important  events  of  the  Middle  Ages.  This 
notation  originated  among  the  Hindus  at  least  as  early  as  a.d.  700. 
It  was  adopted  by  the  Arabs,  and  was  introduced  by  the  Moors  into 
Spain  during  the  twelfth  and  thirteenth  centuries.  Anyone  who  has 
tried  to  multiply  two  numbers  in  the  Roman  notation,  like  MDCC  VII 
by  MCXVIII,  will  realize  the  difficulties  that  surrounded  arithmetical 
operations  before  the  Arabic  system  Avas  taught.  Before  the  introduc- 
tion of  this  system  one  of  the  principal  uses  for  arithmetic  was  the 
determination  of  the  day  of  the  month  on  which  Easter  came.  Roger 
Bacon  in  the  thirteenth  century  urged  the  theologians  *'  to  abound  in 
the  power  of  numbering,"  so  that  they  might  carry  out  these  com- 
putations. Business  computations  were  made  on  the  abacus,  one 
form  of  which  was  a  contrivance  of  wires  and  sliding  balls  on  which 
arithmetical  operations  can  be  performed  with  great  rapidity. 

Though  computation  in  the  decimal  system  was  common  in  Europe 
from  the  thirteenth  century,  the  final  step  in  perfecting  the  notation 
was  not  taken  until  about  1600,  when  Sir  John  Napier  and  others 
made  use  of  the  decimal  ])oint  in  the  modern  sense.  It  was  not  until 
the  V)eginning  of  the  eighteenth  century  that  it  came  into  general  use. 

85.  Percentage.  The  methods  of  algebra  may  be  used 
to  advantage  in  treating  many  problems  in  percentage 
which  are  also  found  in  arithmetic,  and  in  solving  many 
others  which  would  be  difiicult  or  impossible  to  solve  by 
arithmetical  means  alone. 


ORAL  EXERCISES 

1.  What  is  4%  of  60  ?  3.   What  is  3%  of  a:  +  40  ? 

2.  What  is  4%  of  x^  ?        •     4.  What  is  6%  of  4  a;  ~  a  ? 

5.  What  is  the  interest  on  |100  left  for  one  year  at  5%  ? 

6.  What  is  the  interest  on  $100  left  for  6  years  at  5%  ? 

7.  What  is  the  interest  on  $100  left  for  t  years  at  6%  ? 

8.  What  is  the  interest  on  P  dollars  left  for  t  years  at  6%  ? 


JOHN  NAPIER 


EQUATIONS  CONTAINING  FEACTIONS       187 

9.  What  is  the  interest  on  P  dollars  left  for  t  years  at  r  %? 

10.  What  is  the  total  amount  due  at  the  end  of  t  years  if 
P  dollars  are  left  at  r%  ? 

If  two  sums  of  money  are  x  dollars  and  2500  —  x  dollars 
respectively,  express  as  an  equation  the  statement  made  in 
each  of  Exercises  11-14. 

11.  Four  per  cent  of  the  first  sum  equals  $24. 

12.  Eive  per  cent  of  the  first  sum  plus  6%  of  the  second 
sum  equals  $130. 

13.  Six  per  cent  of  the  first  sum  equals  4%  of  the  second. 

14.  Five  per  cent  of  the  first  sum  is  $44  more  than  4<^ 
of  the  second. 

When  P  dollars  are  left  at  simple  interest  for  t  years, 
at  a  yearly  rate  of  r%,  the  total  amount  A  accumulated 
is  given  by  the  following  formula : 

P+(P.  r.  0  =  ^(1+^0=^- 
It  should  be  noted  carefully  that  the  value  of  r  is  a 
fraction  of  which  the  denominator  is  100.     Thus,  if  the 
rate  is  5%,  the  value  of  r  is  -^^-^^  or  .05. 

EXAMPLE 

What  sum  of  money  placed  at  simple  interest  for  2  years  at 
5%  amounts  to  $99  ? 

Solution.    Principal  +  interest  =  99. 


Let 

P  =  the  principal. 

Then 

.05  P  •  2  =  the  interest  for  two  years. 

Therefore 

P  +  .10P  =  .99, 

or 

1.10  P=  99. 

Whence 

P=90. 

Check. 

90  +  9  =  99. 

188  FIRST  COUESE  IK  ALGEBKA 

EXERCISES 

1.  What  sum  of  iiioney  placed  at  interest  for  1   year  at 
6%  amounts  to  $371  ? 

'  2.  What  sum  of  money  placed  at  simple  interest  for  3  years 
■  at  4%  will  amount  to  $476  ? 

3.   In  how  many  years  will  $325,  at  6%  simple  interest, 
gain  $39  ? 

.4.  In  how  many  years  will  $480,  at  (U%  simple  interest, 
gain  $156  ? 

5.  At  what  per  cent  simple  interest  will  $375  gain  $75  in 
4  years  ? 

Solution.    375  x  the  rate  of  interest  x  4  =  75. 

Let  X  =  the  rate  of  interest. 

Then  -  $375.1-  =  the  interest  for  1  year, 

:and  $1500  x  =  the  interest  for  4  years. 

Therefore  1500  x  =  75, 

and  a'=  .>V,  or  -^^^. 

Hence  the  money  is  lent  at  5%. 

6.  At  what  per  cent  simple  interest  will  $725  gain  $145  in 
4  years  ? 

7.  At  what  per  cent  siinplc  interest  will  $250  amount  to 
$317.50  in  6  years  ? 

8.  In  how  many  years  will  $300  double,  itself  at  5% 
simple  interest  ? 

9.  In  how  many  years  will  $500  treble  itself  at  6% 
simple  interest  ? 

10.  A  part  of  $800  is  invested  at  3%,  and  the  remainder  at 
4%.  The  yearly  income  from  the  two  investments  is  $30. 
Eind  each  investment. 

Solution.    One  part  x  .03  +  the  other  part  x  .04  =  30. 

Let  x  =  the  number  of  dollars  invested  at  3%. 

Then  800  —  x  =  the  number  of  dollars  invested  at  4%. 


EQUATIONS  CONTAINma  FRACTIONS       189 

Therefore,  by  the  conditions  of  the  problem, 

.03  X  +  .04  (800  -  x)  =  30. 
Multiplying  each  member  by  100, 

3  2  +  4  (800  -  x)  =  3000. 
Solving,  X  =  200, 

and  800  -  :^  =  600. 

Hence  the  3%  investment  is  $200,  and  the  4%  investment  is  $600. 

Check.  200  600 

.03  .04 

6.00  24.00  $6  4-  $24  =  $30. 

11.  A  part  of  $1500  is  invested  at  5%  aiid  the  remainder 
at  4%.  The  total  annual  income  from  the  two  investments  is 
$69.    Find  the  amount  of  each  investment. 

12.  A  sum  of  money  at  6%  interest  and  a  second  sum  at 
5%  yield  a  total  annual  income  of  $50.  The  first  sum  exceeds 
the  second  by  $100.    Find  each. 

13.  A  4%  investment  yields  annually  just  as  much  as  one 
at  5%.    If  the  sum  of  the  investments  is  $3600,  find  each. 

14.  A  5%  investment  yields  annually  $10  less  than  a  6% 
investment.  If  the  sum  of  the  two  investments  is  $240,  find 
each. 

15.  A  man  invests  part  of  $4300  at  6%  and  the  remainder 
at  5%.  The  investment  at  6%  yields  annually  $1.60  less  than 
the  one  at  5%.    Find  the  sum  invested  at  5%. 

16.  A  man  invests  part  of  $5360  at  5%  and  the  remairidei 
at  6%.  The  yearly  income  from  the  5%  investment  is  $63.40 
more  than  that  from  the  6%  investment.  Find  the  sum 
invested  at  6%. 

17.  A  part  of  $4560  is  invested  at  4%  and  the  remainder 
at  6%.  The  total  yearly  income  is  $202.40.  Find,  the  amount 
invested  at  6%. 


190  riEST  COURSE  IN  ALGEBEA 

86.  Literal  equations.    At  this  poiiit  the  student  should 
review  the  solutions  on  pages  95  and  96. 

EXERCISES 

Solve  for  a?,  and  check  as  directed  by  the  teacher : 

1.  2ax-2a''  =  6a''-2ax.      7.  3(2x  -  a)=  2(x  -  a).  . 

2.  5c^-4.cx  =  c(5x-4.e).      8.  4.(x  -  h)=  2(x -}- b). 

3.  3a^x-4.h^  =  P-2a^x.       9.  4.a(6x-Sh)  =  Sb(S-4:ay 

4.  2  (aj  +  3)  —  4  a  =  6.  10.  ax -\- hx  =  a^ -{-  ah. 

5.  3(:c-l)- 9a  =  6.   ,         11.  cx  +  h''  =  hx-\-hc. 

6.  3(x-2)~lh=^2(x-2).    12.  ax  +  h^  =  a^-hx. 

13.  ^ah-^lbcx-lOhc-'dax^O. 

14.  12-15a+16x  =  20aic. 

15.  aa;  +  Z>aj  4-  ^3?  =  a^  +  ^^  -h  c^- 

16.  2aaj  +  2a  +  ca;  +  c  +  a;+l=0. 

17.  6  aZ>  +  A^a;  +  4  ^2  =  2  aa;  +  3  Z)^  +  2  ah. 

18.  5  aa;  —  5  a^  — 10  (x^  =  3  ac  +  6  Z^c  —  3  CO?. 


cc 


^2 


19.  777  =  a.  24. d  = \-  a. 

LO  XX 

20.  =  5.  25.  7-7  —  4  7^2  =  —I Al 

a?  2  k                     11 

^      'c       3c       5  ^^    2x   ,   a  — 4a?      _              . 

21.  -  +  77-  =  7-  26.  — H — ^-3a=-4. 

X      2x      ^  a             3 

^^8a8a      35a  «H,^.^,            7.7.^ 

22.  ^^  +  —  =  o  +  ^~-  27.  ~  -\--■\^ac  =  U-\^ab^-' 
6x        X        Z  ox                    a       c                                 0 

^^    X      X  .,  3a--4a;.3a  — 2a?       1 

23.  ~-\--  =  a-\-h,  28.  — +  — -j ■  =  7^' 

ah  5  a               4«.           20 

hx 


3a/         2ah\         Jb       a\ 


29.    ^ 


^^    X  —  a       h 

30.  7  =  -- 

X  —  0       a 


EQUATIONS  CONTAINma  FRACTIONS       191 

31  ^  ,  ^         1^  +  ^-0 

h(h-x)      d{b-x)^   2dh  ' 

c  .  k  ^-hc-^-^T^ 

32. 1 = -• 

h  {x  -\-  c)       c(x  —  k)        2kc(x  —  k) 

33.1  +  1-^=1. 
ab  X       X 

87.  Meaning  of  primes  and  subscripts.  Different  but  re- 
lated values  are  often  represented  by  the  same  letter,  with 
smaller  characters  written  at  the  right  and  above  or  below 
the  letter  used ;  as,  y'^  y"^  a^^,  4  x^^  t^j  t^^.  These  are  read 
y  prime,  y  second,  x  sub  zero,  4  x  sub  three,  the  square  of 
t  sub  m,  and  t  sub  w  respectively.  Primes  and  subscripts, 
unlike  exponents,  possess  no  numerical  significance,  and  the 
student  should  carefully  note  that  x^  and  x^  are  as  different 
numerically  as  a  and  b. 

The  notation  just  explained  is  very  convenient  in  physics, 
where  L^  and  L^  may  denote  different  but  related  lengths ; 
IFj  and  W^^  may  represent  two  different  weights ;  and  t^,  t^, 
and  ^2  may  mean  three  unequal  but  related  intervals  of  time. 

Primes  are  cumbersome  and  easily  confused  with  expo- 
nents ;  hence  subscripts  are  preferable. 

The  following  equations  are  taken  from  algebra,  geometry, 
and  physics,  where  it  is  often  necessary  to  express  one  of  the 
quantities  (weight,  time,  distance,  etc.)  in  terms  of  the  others. 

EXERCISES 

1.  Solve  for  R;  K=2  ttRH. 
(Formula  for  curved  surface  of  cylinder.) 

ab 

2.  Solve  for  a  ;  yl  =  —  • 

(Formula  for  area  of  triangle.) 

3.  Solve  for  i2;  C  =  27rR. 

(Formula  for  circumference  of  circle,    tt  =  about  22/7.) 


192  FIEST  COURSE  IN  ALGEBEA 

4.  Solve  for  r  and  t\  d  —  rt. 
(Formula  for  uniform  motion.) 

5.  Solve  for  a  and  ^  ;  •—  =  ^— • 

(Formula  relating  to  the  measurement  of  angles.) 

6.  Solve  fore;  3^=^. 

E 

7.  Solve  for  r;   C  = 

R  -\-r 

(Ohm's  law  for  a  simple  electrical  circuit.) 

E 


8.  Solve  for  r  and  n ;  C 

9.  Solve  for  v  and  n\  C  = 


R  -\-nr 
n  '  e 


R  -\-nr 

10.  Solve  for  F;  C  =  f(F-  32). 

(Formula  for  converting  thermometer  readings   from  one 
scale  (Fahrenheit)  to  another  (Centigrade).) 

W       L 

11.  Solve  for  TT^;  — '  =  -f- 

12.  Solve  for  r  and  t)  A  =.P(1  -f-  rt). 

13.  Solve  for  P,;  ^==^' 

(Formula  relating  to  volume  and  pressure  of  a  gas.) 

14.  Solve  for  n  and  l\  s  =     ^     — ^• 

Z 

15.  Solve  for  a,  Z,  and  t\  s  — —  • 

r  —1 

16.  Solve  for  a ;  -— --  =  —  • 

loO      '    TT 

17.  Solve  for  t^;  V^  =  F„(l  +  .00365 1^). 

18.  Solve  fori,;  ^  =  ^^JL±Mf. 


EQUATIONS  CONTAINING  FRACTIONS       193 

88.  The  lever.  The  figure  given  below  is  a  diagram  of  a 
machine  called  a  lever.  ^C  is  a  stiff  bar  resting  on  a  single 
support  at  B,  This  support  is  called  the  fulcrum  and  AB 
and  BC  are  spoken  of  as  arms  of  the  lever. 

Those  who  have  played  with  a  teeter  board  have  had 
some  experience  with  a  lever,  and  they  have  found  that, 
in  order  to  balance,  the  heavier  of  two  persons  must  sit 
nearer   the   fulcrum      ^  ,, 

than  the  lighter  one  '  ,  ^ 

does. 

In  general,  if  the  lengths  of  the  arms  of  a  lever  are 
?i  and  Zg  and  the  corresponding  weights  are  W^  and  TTg, 
a  balance  results  when 

Thus,  if  AB  =  ^  feet  and  ^C  =  4  feet,  a  boy  at  A  who  weighs 
100  pounds  will  balance  a  boy  at  C  who  weighs  75  pounds ;  for 
3  .  100  =  4  .  75. 

PROBLEMS 

1.  A,  who  is  4  feet  from  the  fulcrum,  balances  B,  who  h 
6  feet  from  it.   A  weighs  96  pounds.    Find  the  weight  of  B. 

2.  A,  who  weighs  100  pounds,  balances  B,  who  weighs  120 
pounds.  B  is  80  inches  from  the  fulcrum.  How  far  from 
it  is  A? 

3.  A,  who  weighs  125  pounds,  balances  B,  who  weighs  100 
pounds.  The  distance  between  them  is  9  feet.  How  far  is 
each  from  the  fulcrum  ? 

4.  A  and  B  together  weigh  210  pounds.  They  balance  when 
A  is  3  feet  9  inches  from  the  fulcrum  and  B  is  5  feet  from  it. 
Find  the  weight  of  each. 

5.  A's  weight  of  110  pounds  is  |-  as  much  as  B's.  They  are 
24  feet  apart.   How  far  from  B  is  the  fulcrum  if  they  balance  ? 


194  FIEST  COUESE  IN  ALGEBEA 

REVIEW  PROBLEMS 

1.  Separate  240  into  two  parts  such  that  their  quotient  is  7. 

2.  Separate  68  into  two  parts  such  that  |-  of  the  greater 
shall  equal  |  of  the  less. 

3.  Separate  |-  into  two  parts  such  that  ^  of  one  part  shall 
equal  -g-  of  the  other. 

4.  Find  two  numbers  whose  sum  is  98  and  such  that 
the  greater  divided  by  the  less  gives  a  partial  quotient  of 
3  and  a  remainder  of  6. 

TT  Dividend      ^    ,.  ,  ^     ,.     ,      Remainder 

Hints.  =  Partial  Quotient  -\ 

Divisor  Divisor 

Let  X  =  the  less  number. 

Then  98  —  x  =  the  greater  number. 

That  is,  ?^:Z^  =  3  +  1 

X  X 

Solve  and  check  as  usual. 

5.  Separate  120  into  two  parts  such  that  one  divided  by 
the  other  gives  a  partial  quotient  of  5  and  a  remainder  of  12. 

6.  The  sum  of  two  numbers  is  1797.  The  greater  divided 
i)j  the  less  gives  a  partial  quotient  of  70  and  a  remainder  of 
22.    Find  the  numbers. 

7.  Separate  ^g^-  into  two  parts  such  that  their  product  is 
greater  by  -g-  than  the  square  of  the  smaller  part. 

8.  The  sum  of  two  numbers  is  16.  Four  times  the  greater 
number  exceeds  50  by  half  as  much  as  28  exceeds  the  less. 
Find  the  numbers. 

9.  A  boy's  age  now  is  -|  of  what  it  will  be  10  years  hence. 
How  old  is  he  now  ? 

10.  One  sixth  of  a  certain  man's  age  12  years  ago  equals  ^ 
of  ]^i^  age  §  years  hence.    What  is  his  age  now  ? 


EQUATIONS  CONTAINING  FRACTIONS       195 

11.  A  collection  of  nickels  and  quarters  contains  60  coins. 
Their  total  value  is  $11.    How  many  are  there  of  each  ? 

12.  Twenty-three  coins,  dimes  and  quarters,  have  the  value 
$3.05.    How  many  are  there  of  each  ? 

13.  The  square  of  half  a  certain  even  number  is  14  less 
than  ^  the  product  of  the  next  two  consecutive  even  numbers. 
Find  the  numbers. 

14.  A  rectangle  is  4  times  as  long  as  it  is  wide.  If  it  were 
4  feet  shorter  and  1^  feet  wider,  its  area  would  be  11  square 
feet  more.    Find  its  length  and  breadth. 

15.  A  rectangle  is  J  as  broad  as  it  is  long.  If  its  length 
were  doubled  and  its  breadth  diminished  by  14,  its  area  would 
be  diminished  by  the  square  of  half  its  breadth.  What  are  its 
dimensions  ? 

16.  It  costs  as  much  to  sod  a  square  piece  of  ground  at 
15  cents  per  square  yard  as  to  fence  it  at  20  cents  per  foot. 
Find  the  side  of  the  square. 

17.  A  rectangular  court  is  twice  as  long  as  it  is  wide.  It 
costs  half  as  much  to  fence  it  at  50  cents  per  yard  as  to  seed 
it  at  15  cents  per  square  yard.    Find  its  dimensions. 

18.  A  rectangular  picture  2^  times  as  long  as  wide  is  sur- 
rounded by  a  frame  2  inches  wide.  The  area  of  the  frame  is 
128  square  inches.    Find  the  dimensions  of  the  picture. 

19.  A  man  bought  apples  at  16  cents  per  dozen.  He  sold  ^ 
of  them  at  the  rate  of  3  for  5  cents,  but  the  rest  were  not  so 
good,  and  he  had  to  sell  them  at  the  rate  of  2  for  3  cents.  He 
made  a  profit  of  48  cents  on  the  entire  transaction.  How  many 
dozen  apples  did  he  buy  ? 

20.  A  can  do  a  piece  of  work  in  2  days,  B  in  3  days,  and  C 
in  4  days.    How  long  will  it  take  them,  working  together  ? 


196  FIEST  COUESE  IN  ALGEBRA 

Solution.  By  the  conditions  of  the  problem  A  does  J  of  the  work 
in  one  day,  B  does  ^  of  the  work  in  one  day,  and  C  does  ^  of  the 
work  in  one  day.  Let  x  represent  the  number  of  days  required  by 
A,  B,  and  C  together  to  do  the  work. 

Then  -  =  the  fractional  part  of  the  work  the  three  together  do 

in  one  day. 

Therefore  -  H ^  -  =  -  • 

2      3      4a: 

■    Solving,  X  =  \  |. 

Check.  -  +  -  +  -  =  —  ,or  —  = 

2      3      4      jl         12      12 

21.  A  can  do  a  piece  of  work  in  6  days,  and  B  in  9  days. 
How  many  days  will  they  require,  working  together  ? 

22.  A  can  do  a  piece  of  work  in  6  days,  B  and  C  each  in 
8  days,  D  in  12  days.  How  many  days  will  they  require, 
working  together  ? 

23.  A  can  do  a  piece  of  work  in  3  days,  B  in  4|  days.  How 
many  days  will  they  require,  working  together  ? 

24.  A  can  do  a  piece  of  work  in  8  days,  A  and  B  working 
together  in  4|  days.    How  long  would  it  take  B  alone  ? 

25.  A  can  do  a  piece  of  work  in  5^  days,  B  in  4|-  days, 
A,  B,  and  C  together  in  1-|  days.  How  long  would  it  take 
C  alone? 

26.  A  can  do  a  piece  of  work  in  8  days.  After  he  has 
worked  3  days,  B  joins  him  and  they  finish  the  work  in  3 
more  days.  How  long  would  it  have  taken  B  to  do  the  work 
alone  ? 

Hint.  What  fractional  part  does  A  do  in  1  day  ?  in  3  days  ?  in  6 
days  ?   What  fractional  part  does  B  do  in  1  day  ?   in  3  days  ? 

27.  A  can  do  a  piece  of  work  in  4  days,  B  in  5  days.  After 
A  has  worked  alone  for  1  day,  B  joins  him  and  they  finish  the 
job  together.    How  much  longer  does  it  take  them  ? 


EQUATIONS  CONTAINING  FRACTIONS       197 


28.  Two  bicyclists  start  at  the  same  time  to  ride  from  A 
to  B,  80  miles  distant.  One  travels  4  miles  an  hour  more 
than  the  other.  The  faster  bicyclist  reaches  B  and  at  once 
returns,  meeting  the  slower  one  at  C,  64  miles  from  A.  Find 
the  rate  of  each. 

Solution.  The  problem  states  that  the  two  travel  at  difPerent 
rates,  that  they  travel  different  distances,  but  that  the  time  is  the 
same  for  each.  Hence  the  equation  must  be  formed  by  expressing 
the  time  ^,  or  d/r^  for  each,  and  equating 
the  two  expressions  for  t.  A 

The  men  together  cover  twice  the  dis- 
tance from  A  to  B,  or  160  miles.  As  the  slower  one  travels  64 
miles,  the  faster  travels  160  —  64,  or  96,  miles.  If  x  equals  the  rate 
of  the  slower  bicyclist  in  miles  per  hour,  we  have : 


C 

— » — 


jB 


rfin  miles 

r  in  miles  per  hour 

-  =  f  in  hours 

r 

Slower  bicyclist 

64 

X 

64 

X 

Faster  bicyclist 

160  -  64  =  96 

:r  +  4 

96 

,      ^  +  4 

Hence 


64 

X 


96 
'a;  +  4' 


Solving,  we  obtain  a;  =  8,  the  rate  of  the  slower  bicyclist  in  miles 
per  hour,  and  a:  +  4  =  12,  the  rate  of  the  faster  bicyclist. 

Check.  -V-  =  8»  and  f  f  =  8. 

29 .  Two  bicyclists,  A  and  B,  start  at  the  same  time  to  ride 
from  X  to  Y,  60  miles  distant.  A  travels  4  miles  per  hour  less 
than  B.  The  latter  reaches  Y  and  at  once  turns  back,  meeting 
A  12  miles  from  Y.   Find  the  rate  of  each. 

30.  A  train  runs  360  miles.  On  the  return  trip  it  increases 
its  rate  by  5  miles  an  hour  and  makes  the  run  in  an  hour  less 
time.    Find  the  rates  going  and  returning. 


198 


FIEST  COUESE  IN  ALGEBRA 


31.  An  automobile  makes  a  run  of  120  miles.  The  chauffeur 
then  increases  the  speed  by  5  miles  an  hour  and  returns  over 
the  same  route  in  4  hours  less  time.  Eind  the  rates  going  and 
returning. 

32.  A  bicyclist  traveling  15  miles  per  hour  was  overtaken 
11^  hours  after  he  started  by  an  automobile  which  left  the 
same  starting  point  3  hours  and  20  minutes  later.  What  was 
the  rate  of  the  automobile  ? 

33.  A  man  travels  at  a  uniform  rate  from  A  to  B,  120  miles 
distant.  He  travels  the  first  70  miles  without  stopping.  The 
remainder  of  the  journey,  including  a  delay  of  2  hours,  requires 
the^  same  time  as  the  first  part.    Eind  his  rate. 

Solution.  By  reading  the  problem  we  discover  that  the  distances 
covered  in  the  first  and  second  portions  of  the  journey  are  different, 
that  the  time  of  travel  is  not  the  same  for  each,  but  that  the  rate 
throughout  is  the  same.  Hence  the  equation  will  be  formed  by 
finding  two  expressions  for  the  rate  r,  or  d/i^  and  setting  them  equal 
to  each  other.  If  x  equals  the  number  of  hours  required  to  travel 
70  miles,  we  have : 


d  in  miles 

t  in  hours 

d/t=r  in  miles 
per  hour 

First  part  of  journey 

70 

X 

70 
X 

Second  part  of  journey 

50 

x-2 

50 

x-2 

Hence 


70 


50 
'x-2' 


Solving,  we  obtain  x  =  7,  the  time  in  hours  occupied  in  traveling 
the  first  70  miles,  and  70  -^-  7  =  10,  the  rate  in  miles  per  hour. 


Check. 


70-^10=7,     and     7-5  =  2. 


34.   A  leaves  a  certain  point  and  walks  at  the  rate  of  S^  miles 
per  hour.    Two  and  a  half  hours  later  B  leaves  the  same  point 


EQUATIOISrS  CONTAmiNG  FRACTIONS       199 

and  drives  in  the  opposite  direction  at  the  rate  of  9^  miles  per 
hour.  How  much  time  must  elapse  after  A  starts  before  they 
will  be  25  miles  apart  ? 

35.  A  and  B  start  at  the  same  time  from  two  points  120 
miles  apart  and  travel  toward  each  other.  A^s  rate  is  2  miles 
per  hour  less  than  B's.  The  latter,  having  been  delayed  2| 
hours  on  the  way,  has  traveled  the  same  distance  as  A  when 
they  meet.    Find  the  rate  of  each. 

36.  A  man  rows  4  miles  per  hour  in  still  water.  He  finds 
that  it  requires  5  hours  to  row  upstream  a  distance  which  he 
can  row  downstream  in  3  hours.    Find  the  rate  of  the  current. 

Hint.  Let  x  =  the  rate  of  the  current.  Then  4  —  x  =  the  rate  up- 
stream, and  4  +  X  =  the  rate  downstream. 

37.  A  man  who  can  row  4|-  miles  per  hour  in  still  water 
rows  up  a  stream  the  rate  of  whose  current  is  2  miles  per 
hour.  After  rowing  back  he  finds  that  the  entire  trip  took 
6  hours.    How  far  upstream  did  he  go  ? 

38.  A  man  who  can  row  4  miles  an  hour  in  still  water  rows 
downstream  and  returns.  The  rate  of  the  current  is  1^  miles 
per  hour,  and  the  time  required  for  the  round  trip  is  15  hours. 
How  many  hours  did  he  take  to  return  ? 

39.  French  scouts  observe  a  German  aeroplane  leave  its 
base,  fly  24  miles  due  west  over  the  French  lines,  and  return 
to  its  starting  point.  They  are  prevented  by  an  encounter  with 
a  German  detachment  from  observing  the  time  of  either  part 
of  the  trip,  but  they  are  able  to  observe  that  the  duration  of 
the  round  trip  was  40  minutes.  At  the  time  there  was  an  east 
wind  blowing,  with  a  velocity  of  15  miles  per  hour.  What  was 
the  velocity  of  the  aeroplane  ? 

40.  A  farmer  pays  $48  for  a  flock  of  sheep.  He  sells  all 
but  two  of  them  for  |50,  and  gains  a  dollar  on  each  sheep 
sold.    How  many  sheep  had  he  ? 


CHAPTER  XVII 


GRAPHICAL  REPRESENTATION 

89.  Temperature  curve.  The  curve  ABCDEF  is  called 
a  graph.  It  was  made  by  a  recording  thermometer.  Such 
an  instrument  is  provided  with  an  arm  carrying  a  pen, 
which  moves  up  as  the  temperature  rises,  and  down  as  it 
falls.  A  clock  movement  runs  a  strip  of  cross-ruled  paper 
under  the  pen,  and  thus  a  continuous  line  is  traced  on  the 
paper.  The  ac- 
companying rec- 
ord extends  from 
2  P.M.  of  a  certain 
Wednesday  until 
10.30  A.M.  of  the 
Friday  following. 
The  numbers  40, 
50,  60,  70,  80, 
and  90  denote 
degrees  Fahrenheit.  There  are  5  spaces  from  50°  to  60°. 
Hence  one  space  corresponds  to  2  degrees.  The  numbers 
2,  4,  6,  8,  and  10  indicate  the  time  of  day.  Whether  this 
is  A.M.  or  P.M.  can  be  determined  by  notin'g  the  position 
of  these  numbers  with  reference  to  the  heavy  curved  lines 
marked  nooi^.  The  point  A  on  the  graph  informs  us  that  at 
2  P.M.  Wednesday  the  temperature  was  80°.  The  point  B 
between  6  p.m.  and  7  p.m.  Wednesday  marks  the  highest 
temperature  recorded. 

200 


GRAPHICAL  REPRESENTATION  201 

The  point  C  tells  us  that  the  temperature  was  about 
65°  at  6  A.M.  Thursday. 

The  preceding  record  was  made  indoors,  and  the  sudden 
fall  from  D  to  E  was  caused  by  the  opening  of  a  door 
leading  into  a  cold  hallway.  The  portion  of  the  graph 
from  D  to  E  shows  that  the  temperature  of  the  room  fell 
approximately  18  degrees  in  about  30  minutes. 

ORAL  EXERCISES 

By  reference  to  the  graph  (p.  200)  answer  the  following : 
1.  With  what  temperature  does  the  record  begin?  end? 
2,.  What  is  the  highest  temperature  recorded  ?  the  lowest  ? 

3.  About  what  time  was  the  highest  temperature  recorded  ? 
the  lowest  ? 

4.  How  often  did  the  instrument  record  a  temperature  of 
82°?  76°?  73°?  68°? 

5.  At  what  times  did  it  record  a  temperature  of  82°  ?  76°  ? 
73°?    68°? 

6.  To  what  practical  use  can  a  graph  such  as  the  one  here 
explained  be  put  ? 

90.  Related  pairs  of  numbers.  A  relation  between  two 
sets  of  numbers  not  necessarily  connected  with  physical 
quantities  such  as  temperature  and  time  can  also  be 
expressed  by  a  graph,  as  will  be  shown  later. 

The  question  What  two  numbers  added  give  four? 
may  be  expressed  by  the  equation  a;  +  y  =  4.  Here  x  and 
y  are  any  two  numbers  whose  sum  is  4.  It  can  be  seen 
by  inspection  that  if  a;  =  2,  ^  =  2  ;  if  a;  =1,  y  =  3  ;  and  if 
2:  =  0,  ^  =  4,  etc.  Or  we  may  assign  to  x  any  value,  say 
—  2  ;  then  the  equation  becomes  —  2  + «/  =  4,  whence  y  =  6. 


202 


FIRST  COUESE  IN  ALGEBEA 


In  this  manner  we  may  obtain  an  unlimited  number  of 
sets  of  related  values  for  x  and  y^  some  of  which  are 
given  in  the  following  table: 

a;  +  ?/  =  4 
ABCDEFGHI 


It  x  = 

2 

1 

0 

3 

4 

-1 

-2 

5 

6 

then  y  = 

2 

3 

4 

1 

0 

5 

6 

-1 

-2 

91.  Definitions  and  assumptions.  In  constructing  the 
graph  of  an  equation  in  two  variables  a  number  of  assump- 
tions must  be  made.  These  assumptions  and  some  neces- 
sary definitions  are  now  stated.    It  is  agreed: 

I.  To  have  two  lines  at  right  angles  to  each  other,  as 
X^OX,  called  the  jr-axis,  and  Y'OY^  called  the  y-axis,  as 
in  the  figure  on  page  203. 

II.  To  have  a  line  of  definite  length  as  a  unit  of  dis- 
tance. Then  the  number  2  will  correspond  to  a  distance 
twice  the  unit  in  length,  the  number  4^  to  a  distance 
of   4i  times  the  unit,  etc. 

III.  That  the  distance  (measured  parallel  to  the  a:-axis) 
from  the  y-axis  to  any  point  in  the  surface  of  the  paper 
be  the  jr-distance  (or  abscissa)  of  that  point,  and  the  dis- 
tance (measured  parallel  to  the  ^-axis)  from  the  a:-axis  to 
the  point  be  the  ^-distance  (or  ordinate)  of  the  point. 

IV.  That  the  ^^-distance  of  a  point  to  the  right  of  the 
^-axis  be  represented  by  a  positive  number,  and  the 
:r-distance  of  a  point  to  the  left  by  a  negative  number; 
also  that  the  ^/-distance  of  a  point  above  the  a;-axis  be  repre- 
sented by  a  positive  number,  and  the  ^-distance  of  a  point 
below  the  a;-axis  by  a  negative  number.  Briefly,  distances 
measured  from  the  axes  to  the  right  or  upward  are  positive^ 
to  the  left  or  downward  are  negative. 


GRAPHICAL  EEPEESENTATION 


203 


V.  That  every  point  in  the  surface  of  the  paper  corre- 
sponds to  a  pair  of  numbers^  one  or  both  of  which  may  be 
positive,  negative,  integral,  or  fractional. 

VI.  That  of  a  given  pair  of  numbers  the  first  be  the 
measure  of  the  a;-distance,  and  the  second  the  measure  of 
the  ^-distance.  Thus  the  point  (2,  6),  or  A  in  the  figure, 
is  the  point  whose  2:-distance  is  2  and  whose  y-distance 


Quadrant  II 

Quadrant  I 

*A      (+'+) 

■'2' 

3 

i 

Xy  + 

-5-4-3 

B* 

12  3 

Vi- 

a?,+ 

Pi 

^   a:,- 

Quadrant  III 

Quadrant  IV 

(-' 

-) 

/ 

(+' 

-) 

is  6.  Again  the  point  (—  4,  —  3),  or  B^  is  the  point  whose 
a;-distance  is  —  4  and  whose  ^/-distance  is  —  3. 

The  point  of  intersection  of  the  axes  is  called  the  origin. 

The  values  of  the  2:-distance  and  the  y-distance  of  a 
point  are  often  called  the  coordinates  of  the  point. 

Though  not  an  absolute  necessity,  cross-ruled  paper  is  a  great 
convenience  in  all  graphical  work.  Excellent  results,  however,  can 
be  obtained  with  ordinary  paper  and  a  rule  marked  in  inches  and 
fractions  of  an  inch  for  measuring  distances.  Hence  the  graphical 
work  which  follows  should  not  be  omitted  even  though  it  is  found 
inconvenient  to  obtain  cross-r^led  paper  for  class  use. 


204 


FIRST  COUESE  IN  ALGEBRA 


1 

T 

- 

r? 

F 

c 

6 

B 

^ 

1 

^ 

X 

/ 

0 

X 

r-' 

— 

,' 

f 

A 

2 

^ 

lii 

L_ 

/ 

EXAMPLE 

Using  ^  inch  for  the  unit  of  measure,  locate  points  corre- 
sponding to  the  pairs  of  related  numbers  which  follow: 
A,  (2,  2);  B,  (1,  3);  C,  (0,  4);  D,  (3, 1);  E,  (4,  0);  F,  (-  1,  5); 
G,  (-  2,  6);  if,  (5,  -  1);  and  /,  (6,  -  2). 

Solution.  According  to 
VI,  page  203,  the  a;-distance 
of  the  point  (2,  2)  is  2  and 
its  ^/-distance  is  2.  Hence  to 
locate  point  A  at  (2,  2)  we 
measure,  according  to  IV, 
two  units  to  the  right  of 
the  origin  on  the  a;-axis  and 
from  that  point  two  units 
upward. 

To    locate    point    F    at 
(—1,    5)    we    measure    on 
the  a:-axis  one  unit  to   the 
left  of  the  origin  and  from  that  point  upward  five  units.    Point 
F  thus  located  corresponds  to  (—1,  5). 

Points  for  the  other  pairs  of  numbers  given  in  the  example  should 
be  located  by  the  pupil.  The  correct  positions  for  these  points  can 
be  seen  in  the  figure. 

Locating  points  as  in  the  example  above  is  called 
plotting  points. 

EXERCISES 

Draw  two  axes  and  plot  the  following  points,  using  ^  inch 
as  the  unit  of  measure. 

1.  (2,3);  (3,1);  (4,2.6);  and  (1.6,  4). 

2.  (4,  -2);  {5.&,  -3);  (6,  -1);  (-1.6,  4);  (-1.8,  2); 
and  (-  3,  -3). 

3.  (3,  0);  (6,  0);  (1,  0);  (0,  0);  (0,  3);  and  (0,  4.5). 

4.  (-  2,  0);  (-  5,  0);  (-  1,  0);  (0,  -  3.5);  and  (0,  --  4X 


GEAPHICAL  REPEESENTATION      205 

5.  If  the  a>distance  of  a  point  is  zero,  where  is  the 
point  located  ?  Where  is  it  located  if  both  of  its  coordi- 
nates are  zero  ? 

92.  The  graph  of  an  equation.  On  page  202  we  computed 
several  sets  of  values  of  x  and  y  for  the  equation  x-^y  =  4:, 
Later  these  points  were  plotted  in  locating  J,  B^  (7,  i>,  E^  F^ 
G,  IT,  and  I  of  the  example  on  the  preceding  page.  It  is 
evident  from  an  inspection  of  their  position  that  a  straight 
line  can  be  made  to  pass  through  all  of  the  points  there 
located.  The  line  drawn  through  these  points  is  said  to 
be  the  graph  of  the  equation  x -\-  y  =  4, 

EXERCISES 

1.  Find  and  tabulate  six  pairs  of  values  of  x  and  y  which 
satisfy  the  equation  2x  -\-  y  =  S.  Draw  two  axes  and,  using 
^  inch  as  the  unit  distance,  plot  each  of  the  points.  Are  the 
six  points  in  a  straight  line  ?  Where  do  all  the  points  lie 
whose  x-  and  y-distances  satisfy  the  equation  2x  -j-  y  =  S? 
What,  then,  is  the  graph  of  the  equation  2x  -\-  y  =  S?  Does 
X  =  3y  y  =  S  satisfy  this  equation  ?  Plot  the  point  (3,  3).  Is 
it  on  the  graph  of  the  equation  ?  If  the  x-  and  ^/-distances  of 
a  point  satisfy  the  equation  2  x  -\-  y  =  S,  where  is  the  point 
located  ?  If  the  x-  and  ^/-distances  of  a  point  do  not  satisfy 
the  equation  2x  -{-  y  =  S,  where  is  the  point  located  ? 

Find  and  tabulate  six  pairs  of  values  for  x  and  y  which  sat- 
isfy each  of  the  following  equations.  Use  numbers  not  greater 
than  10.  Use  at  least  one  negative  value  for  x  and  one  nega- 
tive value  for  y.  Then  plot  the  six  corresponding  points.  Can 
a  straight  line  be  drawn  through  the  six  points  located  for 
each  exercise  ? 

2.  2x-^Sy  =  6.  ^.x-\-y  =  0.  6.2x  =  y, 
3.4rx-3y  =  12.             5.  x  -  y  =:  0,  7.y  =  3x, 


206  FIEST  COUESE  IN  ALGEBEA 

The  preceding  work  should  convince  the  student  that 
the  graph  of  an  equation  of  the  first  degree  in  x  and  y  is 
a  straight  line.  This  fact  can  be  proved,  but  the  student 
would  not  understand  the  proof  were  it  given  at  present. 
Therefore  it  will  be  assumed  that  the  graph  of  every  hnear 
equation  in  two  variables  is  a  straight  line.  And  as  a 
straight  line  is  determined  by  any  two  of  its  points,  in 
graphing  a  linear  equation  in  two  variables  it  will  be 
sufficient  to  plot  any  two  points  whose  x-  and  ^/-distances 
satisfy  the  equation,  and  then  to  draw  through  these  two 
points  a  straight  line.  The  two  points  most  convenient 
to  plot  are  usually  the  two  in  which  the  line  cuts  the 
axes.  Occasionally  these  points  come  very  close  together, 
and  consequently  they  will  not  determine  accurately  the 
position  of  the  line.  In  such  cases  one  should  decide 
on  two  values  of  x  rather  far  apart  (such  as  0  and  5, 
or  0  and  —  5)  and  compute  the  corresponding  values  of 
y.  Two  such  points  will  fix  the  position  of  the  line  with 
sufficient  accuracy. 

If  a  line  goes  through  the  origin  (as  in  Exercise  6  pre- 
ceding), x=0^  y  =  ^  will  do  for  one  point,  but  a  point 
not  on  the  axes  must  be  taken  for  the  second  one. 

The  essentials  of  the  method  of  graphing  a  given  linear 
equation  in  x  and  y  are  illustrated  in  the  following  example. 


EXAMPLE 

Graph  the  equation  2aj  —  3^/  =  —  6.  In  this  equation,  if 
a?  =  0,  2/  =  2 ;  and  if  3/  =  0,  a;  =  —  3.  Here  the  point  (0,  2)  is 
on  the  ?/-axis  in  the  adjacent  figure,  2  units  above  the  origin ; 
and  the  point  (—  3,  0)  is  on  the  cc-axis,  3  units  to  the  left  of 
the  origin.  The  straight  line  through  these  two  points  is  the 
graph  of2aj  —  3^/  =  —  6. 


GRAPHICAL  EEPEESENTATION 

The  necessary  work  may  be  tabulated  as  follows : 
2x— Si/  =  —  6. 


207 


lix  = 

0 

-3 

2 

then  y  = 

2 

0 

H 

-1 

"~ 

1 

7~ 

/ 

x 

^ 

^ 

y 

/^ 

^' 

^ 

^ 

y 

^. 

, 

x^ 

,'b'^ 

*^ 

/' 

«l 

»*, 

X' 

1 

,r^ 

/^ 

0 

V 

Ji 

— ( 

J 

-f 

) 

-4 

J 

^ 

-5 

- 

I 

A. 

^ 

^ 

,y 

X 

-1 

^ 

^ 

/ 

l^ 

y 

c 

Check.  If  an  error  has  been  made  in  obtaining  the  value  of  x  or 
y  from  the  equation,  or  in  plotting  the  values  found,  it  can  be  quickly 
detected  by  plotting  a  third  point,  as  (2,  3^^),  the  values  of  whose 
X-  and  r/-distances  satisfy  the  equation.  If  this  third  point  lies  on  the 
line  determined  by  the  first  two  points,  the  line  has  been  correctly 
located ;  if  it  does  not,  an  error  has  been  made. 

EXERCISES 

Graph  the  following  linear  equations  : 

1.  a^-fy  =  5.  4.  4;r +  3?/  =  12.  1.2x-ij  =  0. 

2.  x-2/  =  4.  5.  5a;-3y  =  15.  %,x~^y  =  0, 

3.  r»  +  2  2/  =  6.  6.  2  a;  -f  3  ?/  =  10.  9.  a^  =  4. 

Hint.  The  equation  x  ==  4  is  equivalent  to  the  equation  a;  +  0  y  =  4. 
This  last  is  satisfied  by  x  =  4  and  any  value  of  y.   Thus  the  pairs  of  values 


208  FIRST  COUESE  IN  ALGEBRA 

(4,  3)  ;  (4,  6) ;  (4,  0) ;  (4,  —  2),  etc.,  satisfy  the  equation  x  +  0?/  =  4. 
Plotting  these  points,  it  is  evident  that  the  required  graph  is  a  line 
parallel  to  the  2/-axis  and  4  units  to  the  right  of  it. 

10.  x  =  -  4.  12.  2/  =  -  3.  14.  x  =  0. 

11.  2/ =  6.  13.  y  =  0.  15.  x  =  6. 

16.  If  a  point  is  on  a  line,  do  the  values  of  its  ir-distance 
and  its  y-distance  satisfy  the  equation  of  the  line  ? 

17.  If  the  values  of  the  a?-distance  and  the  y-distance  of 
a  point  satisfy  the  equation  of  a  line,  is  the  point  located  on 
the  graph  of  the  equation  ? 

18.  Is  the  point  (4,  3)  on  the  line  whose  equation  is 
2^-3^  =  12?   is  (0,6)?   is  (6,0)? 

19.  Can  you  determine  without  reference  to  the  graph  itself 
whether  the  point  (6,  5)  is  on  any  of  the  graphs  of  the  equa- 
tions in  Exercises  1-9  above  ?    If  so,  on  which  does  it  lie  ? 

It  should  now  be  clear  that : 

The  equation  of  a  line  is  satisfied  hy  the  values  of  the 
x-distanee  and  the  y-distance  of  any  point  on  that  line. 

Any  point  the  values  of  whose  x-distance  and  whose  y-distance 
satisfy  the  equation  is  on  the  graph  of  the  equation. 

93.  Graphical  solution  of  linear  equations  in  two  varia- 
bles. If  we  construct  the  graphs  of  the  two  equations 
5a;  +  4^  =  12  and  a;  —  2 ^  =  8  as  indicated  in  the  figure  on 
page  209,  it  is  seen  that  for  the  point  of  intersection  of  the 
graphs  a;  is  4  and  «/  is  —  2.  Since  the  point  (4,  —  2)  is  on 
both  graphs,  these  values  should  satisfy  both  equations. 
Substituting  4  for  x  and  —  2  for  y  in  each  equation,  we 
get  the  identities  20  +(-  8)=  12,  and  4  -(-  4)=  8.  Thus 
the  graphical  solution  of  two  linear  equations  consists  in 
plotting  the  two  equations  and  finding  from  the  graph  the 
value  of  X  and  the  value  of  y  at  the  point  of  intersection. 


GRAPHICAL  EEPRESENTATION 


209 


Since  two  straight  lines  can  intersect  in  hut  one  pointy  there 
can  be  but  one  pair  of  values  of  x  and  y  which  satisfies  a 
pair  of  linear  equations  in  two  variables. 

The  necessary  work  is  tabulated  as  follows : 

5a;  +  4y  =  12  x-1y=^% 


lix^ 

0 

2| 

5 

then  y  — 

3 

0 

-H 

lfx  = 

0 

8 

2 

then  y  — 

-4 

0 

-3 

^ 

Y 

A 

\ 

\ 

\^ 

t 
A 

^^ 

S 

f^ 

1 

V 

« 

x- 

\ 

X' 

V 

^f 

0 

\ 

^ 

V" 

-A 

-r 

-2 

- 

\^ 

; 

( 

f^ 

-^8 

9 

(\. 

^ 

-1 

\ 

Xl 

x' 

\ 

^ 

^ 

.9 

X* 

^ 

V 

4,-| 

P 

3 

- 

-', 

hT^ 

> 

\^ 

^ 

^ 

V 

s 

_ 

-4 

^ 

^ 

s 

xH 

> 

\ 

r^ 

^ 

5- 

\ 

<^ 

X 

s. 

^ 

6- 

N 

^ 

i. 

EXERCISES 


Solve  graphically  the  following  pairs  of  linear  equations  and 
verify  by  substituting  in  each  pair  of  equations  the  x  and  y 
values  of  the  point  of  intersection  as  obtained  from  their  graphs : 


x  +  y  =  5, 
2x  +  9/  =  ^ 
x-hy  =  T, 
X  —  i/  =  3. 


3. 


4. 


3x  +  2/  ==  6, 
2a; -y  =  8. 


5. 


3a;-2?/  =  10, 
x-{-27/=6. 
ic  —  3  y  =  3, 


210  FIRST  COURSE  IN  ALGEBEA 

4:x-y  =  6,  x-27j  =  S,                 x-2y  =  6, 

2x  +  y  =  9.  '  2x-y  =  4..              '2x-4.y  =  9. 

^    2x-\-5y  =  10,  ^^    x-\-2,j  =  4:,  ^^    2x-y  =  ^, 

'  X  -{-  y  =  5.  '  x  -\-  2y  =  6.              '  x  =  Sy. 

94.  Graphical  representation  of  statistics.  Scientific  data 
and  numerical  statistics  from  the  business  world  are  fre- 
quently exhibited  with  striking  clearness  and  brevity  by 
means  of  graphs.  The  form  of  the  graph  obtained  in  any 
case  depends  on  the  character  of  the  relation  between 
the  plotted  numbers.  Sometimes  the  resulting  graph  is  a 
smooth  curve,  and  then  again  it  may  be  an  irregular  con- 
tinuous line  made  up  of  straight  lines  of  various  lengths. 

The  following  graphs  are  types  which  occur  with  increas- 
ing frequency  in  the  magazines  and  in  the  daily  papers. 
Such  graphs  display  data  effectively,  and  inferences  not 
otherwise  apparent  can  often  be  drawn  from  them. 

Biographical  Note.  Rene  Descartes.  One  of  the  two  or  three  most 
important  advances  ever  made  in  mathematics  was  the  discovery  that 
algebraic  equations  could  be  represented  geometrically.  This  great  dis- 
covery was  made  by  Ren^  Descartes  (1696-1650),  the  French  philosopher. 
Though  never  rugged  in  health,  he  took  part  in  several  campaigns  when 
a  young  man,  and  it  is  said  that  during  a  weary  winter  speat  in  camp  in 
Austria  he  first  conceived  the  ideas  that  resulted  in  this  important  work. 
Though  his  writings  read  very  differently  from  a  modern  book  on  the 
same  subject,  yet  he  developed  all  of  the  essentials  of  graphical  repre- 
sentation. He  saw  that  a  letter,  that  is,  a  coordinate,  might  represent 
either  a  positive  or  a  negative  number,  and  so  enforced  upon  mathema- 
ticians the  conviction  that  negative  integers  are  indeed  numbers  and 
that  they  are  useful  in  algebraic  operations.  After  his  time  they  were 
not  usually  ruled  out  as  absurd  or  impossible,  as  was  commonly  the  case 
before.  He  also  introduced  the  modern  exponential  notation,  though  he 
did  not  use  negative  or  fractional  exponents.  To  Descartes  is  due  the 
use  of  the  last  letters  of  the  alphabet  for  the  unknown  and  the  first 
letters  for  the  known  numbers.  Thus  he  would  have  written  the  equation 
x3  —  8  X  +  16  =  40  in  the  form  x^*  —  8  x  +  16  oo  40.  Though  the  sign  = 
was  used  long  before  his  time,  he  did  not  accept  it.  The  asterisk  he  used 
to  indicate  that  a  certain  power  of  the  variable  was  lacking. 


HEME  DESCAKTEti 


GEAPHICAL  EEPKESENTATION 

EXAMPLE  1 


211 


The  census  reports  of  the  United    States    show  that  the 
population  in  millions  for  ten-year  intervals  was  as  follows : 


Year 

1850 

1860 

1870 

1880 

1890 

1900 

1910 

Population  in  millions 

23.2 

31.4 

38.6 

50.2 

62.6 

76.0 

92.0 

The  graphical  representation  of  these  statistics  is  given 
below.  On  the  graph  the  population  is  measured  parallel 
to  the  vertical  axis,  ^o  ^^  ^^  i^^^  representing  5,000,000 
people.  The  ten-year  intervals  are  measured  along  the  hori- 
zontal axis,  with  -^^  of  an  inch  representing  2^  years. 


p 

r 

Tl 

r 

)n 

"~ 

i\t. 

^ 

,/ 

\ 

^l 

/^ 

- 

5. 

KK 

M 

C) 

a 

Y 

.>f 

©^ 

X 

<>^^J 

iX 

& 

^ 

< 

^ 

k^ 

-£ 

0.( 

KK 

.(K 

> 

>^J 

ly 

^ 

6 

> 

<\-^ 

2^ 

i.OV 

\^ 

^ 

U 

^ 

-5 

15. 

m 

.(K 

K)^ 

^ 

^ 

K 

^ 

f^ 

o 

^ 

< 

R 

1 

1 

c 

3 

J 

k 

^ 

1 

1 

T 

Year  "j 

s 

* 

Vl 

QUESTIONS  ON  EXAMPLE  1 

1.  What   population    does    the    graph   indicate    for   1885? 
for  1905? 

2.  What  cause  may  be  assigned  for  the  downward  bend  of 
the  curve  at  1870  ? 


212 


FIRST  COURSE  IN  ALGEBRA 


3.  What  population  does  the  graph  indicate  for  the  United 
States  in  1915  ?   in  1920  ? 

4.  What  causes  might  make  the  population  of  the  United 
States  in  1920  differ  from  the  value  indicated  by  the  curve  ? 


EXAMPLE  2 

Graphical  record  of  atmospheric  pressure  and  wind  velocity 
for  the  Galveston,  Texas,  storm  period  of  August  15-18,  1915. 
The  great  storm  occurred  August  17. 


Wind  velocity 
in  miles  per 
hour 


Barometer 

reading  in 

inches 


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1  1  1  M  1  1  1  M  M  M  1  1  M  M  M  I28. 

QUESTIONS  ON  EXAMPLE  2 

1.  What  was  the  velocity  of  the  wind  at  midnight  August  15  ? 
at  midnight  August  16  ? 

2.  At  what  times  was  the  wind  velocity  40  miles  per  hour  ? 
80  miles  per  hour  ? 

3.  What  was  the  reading  for  barometric  pressure  (height 
of  the  barometer)  at  njidnight  August  15  ?  at  midnight 
August  16  ?    at  2  a.m.  August  17  ? 

4.  What  was  the  lowest  reading  of  the  barometer  and  the 
highest  velocity  of  the  wind  ?    When  did  this  occur  ? 


GEAPHICAL  EEPEESENTATION 


213 


EXAMPLE  3 

In  the  following  graph  the  upper  curve  shows  in  millions 
the  population  of  the  United  States  for  1890,  1900,  and 
1910.  The  middle  and  lower  curves  show  the  average  yearly 
production  and  exportation  respectively  of  wheat  in  millions 
of  bushels  (using  the  average  for  5-year  intervals)  for  the 
years  1891-1915  inclusive.  The  general  tendency  of  produc- 
tion and  exportation  is  made  more  evident  by  plotting  not 
yearly  values  but  averages  for  five-year  intervals,  since  by 
this  method  the  high  or  low  record  for  any  one  year  is  not 
emphasized  unduly. 


Intervals ^ 

1891- 
1895 

1896- 
1900 

1901- 
1905 

1906- 
1910 

1911- 
1915 

Average  yearly  production  .     . 

477 

540 

660 

681 

803 

Average  yearly  exportation  .     . 

167 

197 

140 

116 

•188 

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214 


FIEST  COURSE  IN  ALGEBRA 


QUESTIONS  ON  EXAMPLE  3 

1.  Estimate  the  population  of  the  United  States  for  1892; 
for  1907 ;  for  1912. 

2.  Compare  the  rate  of  increase  in  the  production  of  wheat 
since  1901  with  the  rate  of  increase  of  the  population. 

3.  Does  the  graph  reveal  any  general  tendency  in  the  num- 
ber of  bushels  of  wheat  which  is  exported  ?  What  tendency 
do  you  observe  ? 

4.  Does  the  graph  indicate  that  the  number  of  bushels  ex- 
ported will  increase  to  any  considerable  extent  ?    Explain. 


EXERCISES  ON  THE  GRAPHING  OF  STATISTICS 

1.  The  records  of  the  United  States  Weather  Bureau  of  a 
certain  city  show  hourly  temperature  records  for  a  certain 
day  in  July.  These  records  reported  for  the  hours  ending 
1  A.M.  to  12  P.M.  inclusive,  expressed  in  degrees,  are  as  follows  : 


A.M.,  hour  ending  at 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

Temperature  record  . 

84 

83 

80 

80 

79 

77 

77 

79 

84 

88 

89 

93 

P.M.,  hour  ending  at 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

Temperature  record  . 

98 

100 

88 

72 

81 

86 

84 

82 

80 

75 

74 

74 

Graph  the  foregoing  data. 

A  thunderstorm  occurred  on  the  day  in  which  the  above 
record  was  made.  Fi^om  the  graph  determine  the  time  of  the 
storm.  How  do  you  account  for  the  sudden  fall  in  temperature 
after  2  p.m.  ?  for  the  sudden  rise  in  the  graph  after  4  p.m.  ? 

2.  From  reports  gathered  from  several  cities  it  has  been  esti- 
mated that  for  100  children  in  school  at  the  age  of  8  the  numbers 


GRAPHICAL  EEPRESENTATION 


215 


of  pupils  remaining  in  school  (not  counting  those  eliminated  by 
death)  for  each  year  up  to  and  including  age  18  are  as  follows  : 


Age  of  pupils 

8 

' 

10 

11 

12 

13 

14 

15 

16 

17 
16.5 

18 
8.0 

Number  in  school  . 

100 

100 

100 

98 

97 

88 

70 

47 

30 

Graph  the  foregoing  data.  Let  ^  inch  represent  one  year  on 
the  horizontal  axis  and  ten  pupils  on  the  vertical  axis. 

From  the  graph  determine  the  years  for  which  the  tendency 
to  leave  school  first  becomes  pronounced.  Does  the  graph  show 
a  greater  tendency  on  the  part  of  14-year-old  and  15-year-old 
pupils  to  leave  school  than  on  the  part  of  those  of  12  and  13 
years  ?   How  does  the  graph  show  this  fact  ?   Explain. 

3.  Estimates  worthy  of  consideration  are  here  given  which 
show  a  contrast  in  the  average  weekly  earnings  for  boys  who 
leave  school  at  the  age  of  14  (the  usual  end  of  the  grammar- 
school  period)  with  weekly  earnings  of  boys  who  leave  school 
at  the  age  of  18  (the  usual  end  of  the  high-school  period) 


Age 

Boys  who  leave  school 
at  the  age  of  14  earn 
for  the  years  indicated 
at  the  left  the  weekly 
wages  given  below  : 

Boys  who  leave  school 
at  the  age  of  18  earn 
for  the  years  indicated 
at  the  left  the  weekly 
wages  given  below : 

14 

14.00 

0 

16 

$5.00 

0 

18 

$7.00 

$10.00 

20 

$9.50 

$15.00 

22 

$11.00 

$20.00 

24 

$12.00 

$24.00 

25 

$13.00 

$30.00 

Graph  the  two  sets  of  data  on  the  same  axes. 
Determine  from  the  graph  which  group  of  boys  receives  the 
more  rapid  increase  in  weekly  wage. 


216 


FIEST  COUESE  IN  ALGEBKA 


4.  The  value  in  millions  of  dollars  of  the  merchandise  im- 
ported into  and  exported  from  the  United  States  for  the  years 
1909  to  1916  inclusive  was  as  follows : 


Year 

1909 

1910 

1911 

1912 

1913 

1914 

1915 

1916 

Imports 

1311 

1556 

1527 

1653 

1813 

1893 

1674 

2197 

Exports 

1663 

1744 

2049 

2204 

2465 

2364 

2768 

4333 

Plot  the  foregoing  data,  using  ^  inch  for  one  year  on  the 
horizontal  axis  and  for  250  million  dollars  on  the  vertical  axis. 

How  do  you  account  for  the  decreased  imports  of  1915  ? 
the  decreased  exports  of  1914?  the  increased  exports  of  1915  ? 

5.  The  purchasing  power  of  a  dollar  in  1896  is  compared  to 
the  purchasing  power  of  a  dollar  for  the  years  1897  to  1915, 
respectively,  in  the  following  table : 


Year 

1896 

1897 

1898 

1899 

1900 

1901 

1902 

Cents 

100 

98.4 

95.2 

83.9 

79.5 

74.5 

70.2 

Year 

1903 

1904 

1905 

1906 

1907 

1908 

1909 

Cents 

72.7 

72.7 

68.6 

65.6 

65.6 

66.7 

60 

Year 

1910 

1911 

1912 

1913 

1914 

1915 

1916 

;  Cents 

57.8 

58.8 

56.9 

57.4 

57.7 

56.6 

!  Granh 

the  foi 

^eo-oinfif 

data,  si 

lowing- 

the  cha 

nsre  in  1 

the  pur. 

chasing  power  of  a  dollar  through  the  given  interval. 

What  tendency  in  the  change  in  the  purchasing  power  of  a 
dollar  does  the  graph  indicate  ?  Could  more  or  less  merchandise 
be  bought  for  a  given  sum  in  1900  or  in  1910  ?  Approximately 
how  much  more  ?  A  salary  of  $1500  in  1910  was  equivalent  to 
what  salary  in  1896  ? 


CHAPTER  XVIII 

LINEAR  SYSTEMS 

95.  Definitions.  A  simple  or  linear  equation  in  several 
unknowns  is  one  which  may  be  put  in  such  a  form  that 

(a)  no  unknown  appears  in  any  denominator; 

(6)  only  one  unknown  appears  in  any  term; 

(c?)  only  the  first  power  of  any  unknown  is  involved. 

The  following  equations  are  linear :  2x+3y=l;  4:w+6i;+2w=4; 
the  following  equations  are  not  linear  :  2  x  •{■  %  xy  —  ^  y  =  \\  4/w  —  Q/v 
+  2/m;  =  2  ;   x^  +  3  a;  -  4  ^/  =  (x  +  2)  (2  y  -  3). 

In  a  previous  chapter  we  have  seen  that  a  simple  equa- 
tion  in  one  unknown  has  only  one  root ;  that  is,  the  value 
of  the  unknown  in  such  an  equation  is  a  constant. 

Thus  the  value  of  the  unknown  in  the  equation  3  a;  +  4  =  7  is  the 
number  1,  and  this  is  the  only  root  of  the  equation. 

A  linear  equation  in  two  unknowns  is  satisfied  by  an 
unlimited  number  of  pairs  of  values  for  the  two  unknowns. 
A  single  equation  in  more  than  one  unknown  is  called  an 
indeterminate  equation.  The  unknowns  are  sometimes  called 
variables. 

Thus  the  equation  a;  +  y  =  10  is  satisfied  by  any  pair  of  numbers 
whose  sum  is  10.  Evidently,  if  one  includes  positive,  negative,  and 
fractional  numbers,  there  is  no  limit  to  the  number  of  pairs  whose 
sum  is  10.    (See  page  202.) 

Two  or  more  equations  involving  two  or  more  unknowns 
are  called  a  system  of  equations. 

217 


218  FIEST  COUESE  IN  ALGEBRA 

A  system  of  equations  all  of  which  are  satisfied  by  the 
same  values  of  the  unknowns  is  called  a  simultaneous  system 
of  equations. 

The  two  equations  x  +  y  =  d  and  a;  —  ?/  —  1  form  a  simultaneous 
system.  The  two  numbers  that  satisfy  both  equations  must  be  such 
that  their  sum  is  9  while  their  difference  is  1.  These  conditions  are 
satisfied  by  a;  =  5,  y  =  4. 

A  set  of  values  (one  for  each  unknown)  which  satisfies  an 
equation  in  two  or  more  unknowns  is  sometimes  called  a 
solution  of  the  equation ;  and  a  set  which  satisfies  a  system 
is  often  called  a  solution  of  the  system.  In  this  book,  how- 
ever, the  word  solution  will  be  used  to  denote  the  process 
of  solving  either  a  single  equation  or  a  system.  The  values 
of  the  unknown  which  satisfy  an  equation  in  one  unknown 
will  be  called  roots,  and  a  set  of  values  for  the  unknowns 
satisfying  an  equation  in  two  or  more  unknowns,  or  a 
system  of  such  equations,  will  be  called  a  set  of  roots. 

ORAL  EXERCISES 

In  Exercises  1-4  find  the  value  of  x  corresponding  to  each 
of  the  values  for  i/  indicated  at  the  right : 

1.0^  +  2/ =  10.  y  =  l,5,6. 

2.  x-y  =  l,  ?/  =  4,  10,  0. 

3.  x-\-2y  =  l^.  y=^,^,  8. 
\.  x-2y  =  0.  7/:=l,  4,  -1. 

In  Exercises  5-8  determine  which  of  the  pairs  of  numbers 
written  at  the  right  of  each  equation  satisfies  that  equation. 
(The  first  number  of  a  pair  always  denotes  the  value  of  x.) 

5.  2a;  +  3y  =  10  (2,  2) ;  (1,  3) ;  (8,  -  2). 

6.  3a;^y  =  0  (2,  4);  (4,  2);  (- 1,  3). 


LINEAR  SYSTEMS  219 

l.y^x  =  0  (3,4);  {^,0^)',  (-1,-1). 

8.  3y-6  =  2^  (1,4);  (-3,0);   (-3,-1). 

In  Exercises  9-12  find  two  pairs  of  numbers  which  satisfy 
each  equation,  and  two  other  pairs  which  do  not. 

9.  x-\-2ij  =  l.  11.  2a; +  3?/ =  16. 
10.  cc-3?/  =  5.                            12.  33/  +  4i«=:2. 

96.  Solution  by  addition  and  subtraction.  It  was  shown 
in  the  previous  section  that  there  is  an  unlimited  number 
of  sets  of  roots  of  a  given  linear  equation  in  x  and  y^  and 
that  there  is  also  an  unlimited  number  of  pairs  of  valuen 
of  X  and  y  which  do  not  satisfy  the  equation.  We  now 
proceed  to  give  an  algebraic  method  of  determining  whether 
or  not  there  is  any  common  set  of  rdots  for  two  given  linear 
equations.  It  turns  out  that  there  is  usually  one  and  only 
one  such  set  of  roots.  This  set  may  always  be  found  by  the 
method  illustrated  in  the  following  examples. 

EXAMPLES 

1.  Solve  the  system  {    ^  +  2/  =  ^^  (1) 

^  lx  —  2y  =  l.  (2) 

Solution.    EHminate  x  first,  thus  : 
(l)-(2),  3y  =  3.  (3) 

(3)-^3,  y  =  \.  (4) 

Substituting  1  for  ?/  in  either  (1)  or  (2),  say  in  (1), 

a:  +  1  =  4.  (5) 

Solving  (5),  X  =  3.  (6) 

Check.  Substituting  3  for  x  and  1  for  y  in  (1)  and  (2)  gives  the 
identities  3+1  =  4 

and  3-2=1. 


220  FIEST  COURSE  IN  ALGEBEA 

Solution.  Eliminate  y  first,  thus  : 

(1).2,  10x  +  Qy  =  2.  (3) 

(2)  .  3,  9x-Qy  =  36.  (4) 

(3)  +  (4),  19  x  =  38.  (5) 
(5)-i-19,  x  =  2.  (6) 

Substituting  2  for  x  in  (2),  Q-2y  =  12.  (7) 

Solving  (7),  ?/  =  -  3.  (8) 

Check.    Substituting  2  for  x  and  —  3  for  y  in  (1)  and  (2)  gives 

10  -  9  =  1, 
and  6  +  6  =  12. 

Either  x  ov  y  could  have  been  ehminated  first.  The 
multiphers  necessary  to » eliminate  x  are  3  and  5,  while  the 
multipliers  necessary  to  eliminate  y  are  the  more  convenient 
numbers  2  and  3. 

When  the  notation  (3)  —  (4)  is  used  in  a  solution,  it  indicates  the 
subtraction  of  the  first  member  of  equation  (4)  from  the  first  member 
of  equation  (3),  the  subtraction  of  the  second  member  of  equation  (4) 
from  the  second  member  of  equation  (3),  and  the  writing  of  the  two 
results  as  an  equation.  The  process  of  adding  the  corresponding 
members  of  the  two  equations  is  indicated  by  writing  (3)  +  (4). 

The  notation  (3)  •  6  indicates  that  both  members  of  equation  (3) 
are  multiplied  by  6,  and  (3)  -^  6  indicates  that  both  members  of 
equation  (3)  are  divided  by  6. 

With  the  meanings  just  explained  it  is  customary  to  speak  of  the 
addition  or  the  subtraction  of  two  equations  and  of  the  multiplica- 
tion or  division  of  an  equation  by  a  number. 

The  method  of  the  preceding  solutions  is  stated  in  the 
Rule.    If  necessary^  multiply  the  first  equation  by  a  number 
and  the  second  equation  by  another  number^  such  that  the  coeffi- 
cients of  the  same  unknown  in  each  of  the  resulting  equations 
will  be  numerically  equal. 


LINEAK  SYSTEMS  221 

If  these  coefficients  have  like  signs^  subtract  one  equation 
from  the  other ;  if  they  have  unlike  signs^  add.  Then  solve 
the  equation  thus  obtained. 

Substitute  the  value  just  founds  in  the  simplest  of  the  preced- 
ing equations  which  contains  both  unknowns^  and  solve  for  the 
other  unknown. 

Check,  Substitute  for  each  variable  in  the  original  equations 
its  value  as  found  by  the  rule.  If  the  resulting  equations  are 
not  obvious  identities^  simplify  them  until  they  become  such. 

An  attempt  to  solve  by  the  rule  the  pair 

r3a:-6y  =  40,  (1) 

1    x-2y  =  %,  (2) 

gives  3  a;  -  6  y  =  40,  (3) 

3a;-6y  =  24.  (4) 

(3)  -  (4),  0  =  16,  which  is  false. 

This  result  indicates  that  (1)  and  (2)  do  not  form  a  simultaneous 
system,  but  are  incompatible  equations. 

The  graphs  of  a  pair  of  incompatible  linear  equations  are  parallel 
lines  (see  Exercise  10,  p.  210).  ^     _ 

An  attempt  to  solve  by  the  rule  the  system  <         i  «     _  ol  gi^^ 

0  =  0.  Here  the  second  equation  divided  by  3  gives  the  first.  There- 
fore any  set  of  roots  of  the  first  is  a  set  of  the  second.  If  we  choose 
to  regard  the  two  equations  as  really  different,  which  is  not  at  all 
necessary,  we  say  that  they  have  an  infinite  (unlimited)  number  of 
sets  of  roots.  Two  or  more  equations  having  this  property  constitute 
an  indeterminate  system. 

EXERCISES 

Solve  the  following  systems  and  check  results : 

X  —  y  =1.  X  —  1/  =  b.  X  —  y  =  o. 

^^  x  +  y  =  7,  ^x  +  y  =  5,  ^x-i-2y  =  7, 

'  03  — y  =  l.  '  X  —  Sy  =  1.  '  5x  —  2y=ll, 


222  WEST  COURSE  IN  ALGEBEA 

22/  +  aJ  =  4,  5m-3n  =  0,  3x-4.y  =  U, 

'•  3y-x  =  21.  15m-{-12n  =  75.  3i/-4:X^-U. 

7r-s  =  2,  x-6y  =  7,  4:X-\-Sy  =  5, 

^'  6r-s  =  S,  122/-aj=:-l.  9y-8a^  =  0. 

Sx-y  =  3,  2x  +  25y  =  70,  6u-{-Sv  =  26, 

5x  +  2y=16,  3a;  =  10y4-10.         V5i*-3i;  =  70. 

10.  ^-+/= J  15.  ^^+/=f;    20.  ""+;^==y'' 

x  —  2 y  =  5.  J)  —  4: q  =14:.  3y  —  4:X  =  4.  . 

3a:+102/  =  4.  3a;-3?/  =  10.  9^  +  153/=:lll. 

97.  Solution  by  substitution.  The  method  of  solving  a 
system  of  two  linear  equations  by  substitution  is  illus- 
trated in  the  following  example. 


EXAMPLE 

CI  1      XT,  .        (4:X-12y  =  U,  (1) 

Solve  the  system  {^^^^^^^^3;  J^j 

Solution.    From  (1),  4  a;  =  12  y  +  44,  (3) 

or  *    '       *    a:  =  3  ?/  + 11.  (4) 

Substituting  3  2/  +  11  for  a;  in  (2), 

8(32^  +  ll)  +  ll.y  =  18.  (5) 

Simphfying,        24  y  +  88  +  11  y  =  18.  (6) 

Collecting,  35y=-70,  (7) 

or  y=-2,  (8) 

Substituting  —  2  for  y  in  (4), 

a:  =-6 +11  =  5.  (9) 

Check.    Substituting  5  for  x  and  —  2  for  y  in  (1)  and  (2)  gives 
20  +  24  =  44, 
and  40  -  22  =  18. 


LINEAE  SYSTEMS  223 

The  method  of  the  preceding  solution  is  stated  in  the 

Rule.  Solve  either  equation  for  one  unknown  in  terms  of 
the  other. 

Substitute  this  value  in  place  of  the  unknown  in  the  equation 
from  which  it  was  not  obtained  and  solve  the  resulting  equation. 

Substitute  the  defiiiite  value  just  founds  in  the  simplest  of 
the  preceding  equations  which  contains  both  unknowns^  and 
solve,  thus  obtaining  a  definite  value  for  the  other  unknown. 

Check.    See  page  221. 

The  method  of  substitution  emphasizes  the  fact  that 
the  values  of  x  and  7/  which  are  sought  are  the  same  in 
both  equations.  Hence  an  expression  for  an  unknown 
obtained  from  one  equation  is  substituted  for  that  un- 
known in  the  other  equation.  This  method  is  useful  when 
one  of  the  unknowns  can  be  expressed  in  terms  of  the 
other  without  using  fractions  or  when  simple  fractions 
only  are  involved. 

EXERCISES 

Solve  by  the  method  of  substitution  and  check  results : 
x-2y  =  S,  Sx  =62j~-S, 


1. 
2. 
3. 


Sx-{-2y  =  S.  ?/-f4a;  =  5. 

^_22/=-l,  5x-M07/  =  25, 

4:x-y=10.  '  5x-Sy=9. 

14m-2/i=l,  20y'-Sz=l, 

n  —  6m  =  0.  '  z  —  61/  =  0. 

6a;4-10y  =  42,  ^^    3a: -f  12  =  3  +  2/, 

22/==3x.  '  y  =  X  -{-1. 

5x-]-Sy  =  -l,  lS-\-2k  =  h, 

2x-6y  =  S.  k-\-k=-9, 

^-hy=7,  2aj  =  4?/-f6, 

2x-}-Sy=lT.  7x-\-Sy  =  4., 


224  FIEST  COURSE  IN  ALGEBEA 

98.  Simultaneous  equations  containing  fractions.  The 
method  of  solving  a  system  of  two  linear  equations  con- 
taining fractions  is  illustrated  by  the' following  example. 


(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

The  system  (4)  and  (6)  can  now  be  solved  by  addition  and 
subtraction. 

As  in  the  foregoing  solution,  it  is  usually  best  to  clear 
the  equations  of  fractions  and  write  them  in  the  form  of 
(4)  and  (6)  before  attempting  to  eliminate  one  of  the 
unknowns.  Equations  (4)  and  (6)  are  each  in  what 
is  called  the  general  form  of  a  linear  equation  in  two 
unknowns.  This  form  is  represented  for  all  such  equa- 
tions by  ax-\-'by  =  (?,  where  a,  5,  and  c  denote  numbers, 
or  known  literal  expressions. 

EXERCISES 

Solve  the  following  systems  and  check  results : 


EXAMPLE 

3        6~   2 
2x      y       7 
S   "4"^  12* 

Solve  the  system 

Solution.   (l)-6, 

16  a;  -  5  =  9  ?/. 

Transposing  in  (3), 

16  a: -9?/ =  5, 

(2).  12, 

8  a:  =  3  2/  +  7. 

Transposing  in  (5), 

8a;- 

-33^  =  7. 

,       .   2y      26 

^^     ,     y         rr 

3y      18 

4  X  -f  -TT-  =  -—  > 

-T"  +  .  =  7, 

3        3 

6       4       ' 

5        5 

1. 

2*  ^ 

3. 

Q                ^^                  A 

2^-^  =  3 
3       8      "*• 

7,  +  ^=_16 

LINEAE  SYSTEMS  225 

.4  a; +  .9  2/ =  5.7,  i   .  i_5  " 

4-   _^-i.  13  ^   y~^' 

5.   3       2        6  Hint.  Solve  without  clearing  of 

2  7/  —  3  ic  =  0.  fractions,  using  addition  and  sub- 
traction. 
X       y  _      7 

^    3"^4"""8'  1,1       4 

y       ^_11  14    ^       ^       1^ 

3""4~12'  '  5^_?_19. 

.04x  +  .32/=l,  ^ 

.5 ^-.25 2/ =  4.5.  <5  ,   '^   ,  3      A 

7.      4,,  =14,  15.^      ^ 

8.  2a;+7  .  =  — . 

— 2 3/ =  4-  a;      y        3 

2      2/  _  1  -  +  12  =  17, 

a;-2    J/  +  l_l  --3y  =  0. 


5  2  2 


aj 


2ic  +  6  .  2/  +  4:      ,  0^-3/^  25      a;  +  y 
+  — F-  =  ^»            _        2  6  3 


5        '       5 


17. 


aj  +  2/       Q      x  —  ^J  x-2  _y-^10      10  —  x 

n    ~2-~'=^~3-'  18.       ^              4      -      3      ' 

'  x-y      xj-y__  y-h2  _2x-\~y  ^x-hlS 

"~4~"  +  ""'3"' ""  3             16              8      * 

---  =  1,  __  +  8~2a.--^-, 

•i±^^_3,_^  +  4  =  0.  -2,-^^  =  3.4-4. 


226  FIEST  COUESE  IN  ALGEBRA 

2 3 

^^x-y      20^4-32/ -14-^'  (^) 


2x-^-^- 


(2) 


4      4 
Hints.   Multiplying  (1)  by  (x  —  y)  (2  x  +  3  y  —  14), 

2  (2  X  +  3  2/  -  14)  -  3  (X  -  2/)  =  0,  (3) 

or  x  +  92/  =  28.  (4) 

(2).  4,  8x-2/  =  5.  (5) 

Now  solve  equations  (4)  and  (5)  by  substitution  and  check  as  usual. 


4g;  +  y  _  2  5a;  +  3?/  —  5 

+  ^~^'  0.5.  3 

102/_71  25  a; 


21.  'n/      ''  35.  3  ~'''^' 


3  3  10a; +  2y 


=  1. 


22    ^^^^         '  26     ^  +  ^       ^-^ 


4a;  —  3y-f-l       4  r  +  s  +  1 

3a;-4.y  +  2_,  2 


23,  ^-y  27    ^+^  +  2      3a;  +  5y-5 

a;  3*  4"^2~ 

aj_3c«-6  4  3 


24.  ^i^      y-^  28."-/       ^~^, 

a;  +  y  -  5  _  ^  5  7 


x-y-\-\         *  22/-5       2a;-39 

In  the  following  problems  the  student  should  state  two 
equations  in  two  unknowns.  Instead  of  using  x  and  y, 
the  jiTBt  letter  of  the  word  denoting  an  unknown  should 
be  used  to  represent  that  unknown.  Thus  in  Problem  5, 
p.  227,  n  should  represent  the  number  of  nickels  and  q 
the   number   of   quarters.     The   plan    here    suggested   is 


LINEAR  SYSTEMS  227 

desirable  for  many  reasons,  and  should  be  followed  in  all 
problems  containing  two  or  more  unknowns  unless  the 
words  denoting  two  of  the  unknowns  begin  with  the 
same   letter. 

In  solving  problems  like  1-12  on  pages  46-47,  there 
are  really  two  unknowns  involved ;  but  one  of  the  equa- 
tions to  which  each  of  those  problems  leads  is  so  simple 
that  what  amounts  to  the  method  of  substitution  was  em- 
ployed by  expressing  one  unknown  in  terms  of  the  other. 

PROBLEMS 

1.  The  difference  of  two  numbers  is  20  and  their  sum  is  36. 
Find  the  numbers. 

2.  The  quotient  of  two  numbers  is  12  and  their  sum  is  39. 
Find  the  numbers. 

3.  Find  two  numbers  whose  difference  is  28  and  whose 
quotient  is  5. 

4.  The  value  of  a  certain  fraction  is  |.  If  6  be  added  to 
the  numerator  and  12  to  the  denominator,  the  value  of  the 
resulting  fraction  is  |.    Find  the  fraction. 

5.  A  collection  of  nickels  and  quarters,  containing  63  coins, 
amounted  to  $8.35.    How  many  coins  of  each  kind  were  there  ? 

6.  If  1^  be  subtracted  from  the  numerator  and  ^  added  to 
the  denominator  of  a  certain  fraction,  the  value  of  the  result- 
ing fraction  is  j.  The  sum  of  the  numerator  and  the  denomi- 
nator of  the  original  fraction  is  11.    Find  the  fraction. 

7.  The  difference  between  the  numerator  and  the  denomi- 
nator of  a  certain  proper  fraction  is  11.  If  |^  be  added  to  the 
numerator  and  f  be  taken  from  the  denominator,  the  value  of 
the  resulting  fraction  is  -^q.    Find  the  fraction. 


228  FIRST  COUESE  IN  ALGEBE-A 

8.  Two  weights  balance  when  one  is  12  inches  and  the 
other  8  inches  from  the  fulcrum.  If  the  second  weight  increased 
by  8  pounds  is  placed  6  inches  from  the  fulcrum,  the  balance 
is  maintained.    Eind  the  two  weights. 

9.  Two  weights  balance  when  one  is  10  inches  and  the 
other  8  inches  from  the  fulcrum.  If  the  first  weight  is  de- 
creased by  4  pounds,  the  other  weight  must  be  moved  2  inches 
nearer  the  fulcrum  to  balance.    Find  the  weights. 

10.  In  20  years  A  will  be  twice  as  old  as  B.  Ten  years  ago 
A  was  8  times  as  old  as  B.    Find  the  age  of  each  now. 

11.  The  perimeter  of  a  rectangle  is  184  feet  and  the  length 
is  8  feet  more  than  twice  the  width.  Find  the  dimensions  of 
the  rectangle. 

12.  A  part  of  $1500  is  invested  at  6%  and  the  remainder 
at  5%.  The  combined  yearly  income  is  $81.  Find  the  number 
of  dollars  in  each  investment. 

13.  A  part  of  $3000  is  invested  at  4|-%  and  the  remainder 
at  3|^%.  The  annual  income  from  the  4|^%  investment  is  $3 
less  than  double  the  annual  income  from  the  3|-%  investment. 
Find  the  number  of  dollars  in  each  investment. 

14.  A  part  of  $5000  is  invested  at  4%  and  the  remainder 
at  6%.  The  4%  investment  yields  $16  less  in  4  years  than 
the  one  at  6%  does  in  3  years.  Find  the  number  of  dollars  in 
each  investment. 

15.  Five  rubles  are  worth  5  cents  less  than  10  marks,  and 
12  marks  are  worth  4  rubles  and  a  dollar.  Find  the  value  of 
a  ruble  and  of  a  mark  in  cents. 

16.  During  the  war  the  value  of  marks  in  New  York  fell  so 
much  that  on  Jan.  1,  1916,  6  marks  were  worth  exactly  as 
much  as  5  marks  were  worth  on  Aug.  1,  1914.  Between  the 
two  dates  mentioned  the  value  of  10  marks  decreased  by  40 
cents.    Find  the  value  of  a  mark,  in  cents,  at  each  date. 


LINEAE  SYSTEMS  229 

17.  The  sum  of  the  two  digits  of  a  two-digit  number  is  9.  If 
45  be  subtracted  from  the  number,  the  result  will  be  expressed 
by  the  digits  in  reverse  order.    Eind  the  number. 

Solution.    Let  t  =  the  digit  in  tens'  place, 

and  u  =  the  digit  in  units'  place. 

Then  <  +  w  =  9.  (1) 

But  t  standing  in  tens'  place  has  its  numerical  value  multiplied  by  10. 
Therefore  the  number  is  represented  by  the  binomial  10  t  +  u,  and 
the  number  formed  by  the  digits  in  reverse  order  is  represented  by 
the  binomial  10  u  +  t. 


Hence                    10  t  +  u  —  4:6  =10u  +  t. 

(2) 

Simplifying  (2),                 t  —  u  =  5. 

(3) 

Solving  (1)  and  (3),                 t  =  7,  and  m  =  2. 

Hence  the  number  is  72. 

Check.    7  +  2  =  9,  72  -  45  =  27. 

18.  The  sum  of  the  digits  of  a  two-digit  number  is  10.  If  54 
be  added  to  the  number,  the  result  is  expressed  by  the  digits 
in  reverse  order.    Find  the  number. 

19.  The  tens'  digit  of  a  two-digit  number  is  half  the  units' 
digit.  If  36  be  added  to  the  number,  the  result  is  expressed 
by  the  digits  in  reverse  order.    Eind  the  number. 

20.  If  a  two-digit  number  be  divided  by  the  sum  of  its 
digits,  the  quotient  is  4.  Twice  the  given  number  is  9  greater 
than  the  number  expressed  by  the  same  digits  in  reverse  order. 
Eind  the  number. 

21.  If  a  two-digit  number  be  increased  by  4  and  then  the 
result  be  divided  by  the  sum  of  its  digits,  the  quotient  is  3. 
If  twice  th^  number  be  divided  by  the  tens'  digit,  the  quotient 
is  29.    Eind  the  number. 

22.  If  a  two-digit  number  be  divided  by  the  sum  of  its 
digits,  the  quotient  is  7.  If  the  number  formed  by  the  digits 
in  reverse  order  be  divided  by  3  plus  the  sum  of  the  digits, 
the  quotient  is  3.    Eind  the  number. 


230  FIRST  COUESE  IN  ALGEBRA 

The  reciprocal  of  a  number  is  a  fraction  of  which  1  is  the 

numerator  and  the  number  itself  is  the  denominator.    Thus  A 
1  .  ^ 

and  -  are  the  reciprocals  of  2  and  a  respectively. 

23.  What  is  the  reciprocal  of  6  ?  1  al  ^?  ^?  2i  ?  3|  ? 

24.  The  sum  of  the  reciprocals  of  two  numbers  is  W^  and 
the  difference  of  their  reciprocals  is  ■^-^,    Find  the  numbers. 

25.  The  difference  of  the  reciprocals  of  two  numbers  is  2^* 
The  quotient  of  the  greater  number  divided  by  the  less  is  ^-1. 
Find  the  numbers. 

26.  If  5  grams  be  taken  from  one  pan  of  a  balance  and 
placed  in  the  other,  the  sum  of  the  weights  in  the  first  will 
be  |-  the  sum  of  those  in  the  second.  But  if  15  grams  be 
taken  from  the  second  and  placed  in  the  first,  the  weights  in 
the  two  pans  will  then  balance.  Find  the  weight  in  each  pan 
at  first. 

27.  A  gives  B  |30 ;  then  B  has  twice  as  much  money  as  A. 
B  then  gives  A  $150  and  has  left  \  as  much  as  A.  How  many 
dollars  had  each  at  first  ? 

28.  The  circumference  of  the  fore  wheel  of  a  carriage  is 
\\  feet  less  than  that  of  the  rear  wheel.  The  fore  wheel  makes 
as  many  revolutions  in  going  286  feet  as  the  rear  wheel  does 
in  going  325  feet.    Find  the  circumference  of  each  wheel. 

29.  A  and  B  together  can  do  a  piece  of  work  in  7^  days. 
They  work  together  for  3  days,  and  A  finishes  the  job  by  him- 
self in  7  days.    How  many  days  would  each  require  alone  ? 

30.  A  man  rows  10  miles  downstream  in  2  hours  and  returns 
in  4  hours.  Find  the  rate  of  the  river  and  his  rate  in  still  water. 

Hint.  Let  x  —  the  man's  rate  in  still  water  in  miles  an  hour,  and 
y  =  the  rate  of  the  river  in  miles  an  hour.  Then  his  rate  downstream 
is  X  +  2/  miles  an  hour,  and  upstream  x  —  y  miles  an  hour. 


LIKEAE  SYSTEMS  231 

31.  A  boat  goes  downstream  45  miles  in  3  hours  and  up- 
stream 15  miles  in  3  hours.  Find  its  rate  in  still  water  and 
tlie  rate  of  the  current. 

32.  The  rate  of  a  boat  in  still  water  is  7^  miles  an  hour.  It 
goes  down  the  river  from  A  to  B  in  12  hours.  It  returns  one 
half  the  distance  from  B  to  A  in  9  hours.  Find  the  rate  of 
the  river  and  the  distance  from  B  to  A. 

33.  A  boat  which  runs  12  miles  an  hour  in  still  water  goes 
downstream  from  A  to  C  in  7  hours.  It  returns  upstream  to 
B,  36  miles  below  A,  in  5  hours.  Find  the  distance  from  A  to 
C  and  the  rate  of  the  stream. 

99.  Literal  equations  in  two  unknowns.  Linear  systems 
in  which  the  unknowns  have  literal  coefficients  are  solved 
by  the  method  of  addition  and  subtraction. 

EXERCISES 

In  the  following  exercises  consider  a,  h,  c,  and  d,  as  known 
numbers ;  solve  for  the  other  letters  involved  and  check. 

X  -\-i/  =  Sa,  3x  —  i/  =  10b, 

'  X  —  y  =  a.  *4x-[-9?/  =  3Z>. 

5a;-f42/  =  17a,  5x  —  4y  =  10a  —  4, 

*  8  a;  -h  2/  =  11  ^-  '  x  —  2  mj  =  0. 

Sp-^4:q  — a=p  — q-\-22a, 
p  •\-  a  —  q  =  Sj) 


6. 


3x-\-4:y=6c,  8. 

-  _  i^  =  2. 

c       4c         *  9. 


2x      2y      6a 
7.  10. 

y  .  ^ 

2  +  2  =  «- 


2q 

+  8  a. 

7.5 

m  +  3: 

n  = 

6  b, 

.25 

m  +  .5 

n  — 

:0. 

3x 

-3y- 

-4, 

a  =  X  ■ 

-h, 

X  -\-  y  -\-  a 

+  ^ 

=  2  (a 

^b). 

X 

2a 

a 

=  — 

10, 

3x 
a 

4a 

3 
2 

232  FIKST  COtJESE  IK  ALGEBEA 

3  2 
12.     2     ,     ^     ^    ^       '  X      y 

a  .  h 
^3^  coj  +  y  =  3,                                18.  "^      ^ 

'  c  (?/  —  3)  =  cc.  'a       ^  _   , 

^^*  30^-2^/  =  9a -8  5.  19.  ^^  +  ^2/  =  ^  +  ^ 

ax  —  hy  =  a  —  h. 


2x  —  ay  =  h,  ^^    (^x  +  hy  =  c, 

Sx  —  by  =  4ia.  '  hx  -\-  Jcy  =  m. 


GENERAL  ORAL  EXERCISES 

1.  If  one  book  costs  a  dollars,  what  will  c  -\-  cl  books  cost  ? 

2.  If  (X  books  cost  b  dollars,  what  will  one  book  cost? 
c  books  ? 

3.  What  is  the  perimeter  of  a  rectangle  whose  length  is  a 
and  whose  width  is  Z>  ?    What  is  its  area  ? 

4.  What  is  the  perimeter  of  a  rectangle  whose  length  is  4  a 
and  whose  width  is  5  ?    What  is  its  area  ? 

5.  The  base  of  a  triangle  is  a  -{-  b.    The  altitude  is  a  —  b. 
Find  the  area. 

6.  The  base  and  the  altitude  of  a  triangle  are  each  equal 
to  X  ■—2y.    Find  the  area. 

7.  The  area  of  a  triangle  is  k.    The  base  is  b.    Find  the 
altitude. 

8.  If  ic  denotes  A's  age  now,  what  does  x  —  S  denote  ? 
X  -\-  5?    What  does  the  equation  x  -\-  5  =  2(x  —  8)  signify  ? 


LINEAE  SYSTEMS  233 

9.  Forty  men  pay  d  dollars  each  as  dues  to  a  society.  The 
expenses  of  the  society  are  n  dollars.  How  many  dollars  are 
left  in  the  treasury  ?  How  would  you  interpret  the  result  if 
n  is  greater  than  40  c?  ? 

10.  If  10  apples  can  be  bought  for  x  cents,  how  many  can 
be  bought  for  y  cents  ? 

11.  If  a  apples  can  be  bought  for  25  cents,  how  many  can 
be  bought  for  c  cents  ? 

12.  A  farmer  has  grain  enough  to  last  one  horse  d  days. 
How  long  will  this  grain  last  k  horses  ? 

13.  A  farmer  has  grain  enough  to  last  h  horses  d  days. 
How  long  will  it  last  k  horses  ? 

GENERAL  PROBLEMS 

1.  The  altitude  of  a  triangle  is  a  inches  and  the  base  is 
10  inches.  If  2  inches  be  taken  from  the  altitude,  by  how 
much  must  the  base  be  increased  so  that  the  area  will  be  the 
same  as  before  ? 


Hint.   Let  x  =  the  increase  of  the  base  in  inches. 

10a_  (a-2)(10  +  x) 

2     ~  2 


Then 


2.  The  altitude  of  a  triangle  is  a  feet,  the  base  is  h  feet. 
The  altitude  is  increased  h  feet  and  the  base  decreased  so 
that  the  area  is  the  same  as  before.  How  many  feet  are  taken 
from  the  base  ? 

3.  The  sum  of  two  numbers  is  s  and  their  difference  is  d. 
Find  the  numbers. 

4.  The  first  of  two  numbers  is  a  times  the  second,  and  the 
first  minus  the  second  equals  h,   Eind  the  numbers. 

5.  The  sum  of  two  numbers  is  h,  and  the  quotient  of  the 
first  divided  by  the  second  equals  a.   Eiud  the  numbers. 


234  FIRST  COUESE  IN  ALGEBRA 

6.  If  ci^  be  added  to  the  numerator  of  a  certain  fraction, 

the  value  of  the  resulting  fraction  is  2.    If  b  be  added  to  the 

denominator,  the  value  of  the  resulting  fraction  is  1.    Find 

the  fraction. 

Hint.   Let  -  =  the  fraction.    Then  =  2,  and  =  1. 

d  d  d  +  6 

7.  If  the  numerator  of  a  certain  fraction  be  increased  by 
•1,  the  value  of  the  resulting  fraction  is  x.  If  the  denominator 
of  the  fraction  be  decreased  by  2,  the  value  of  the  resulting 
fraction  is  y.    Find  the  numerator  and  the  denominator. 

8.  The  value  of  a  certain  fraction  is  b.  If  2  be  added  to 
the  numerator,  the  value  of  the  resulting  fraction  is  c.  Find 
the  numerator  and  the  denominator. 

9.  A  boy  who  weighs  a  pounds  and  one  who  weighs  b 
pounds  balance  at  the  opposite  ends  of  a  teeter  board  whose 
length  is  I  feet.    How  far  is  the  fulcrum  from  each  boy  ? 

10.  A  certain  number  of  books  at  80  cents  each  and  another 
number  at  fl.lO  each  cost  together  h  dollars.  If  the  prices  of 
the  books  had  been  interchanged,  the  total  cost  would  have 
been  k  dollars.    Find  the  number  of  each  kind. 

11.  Two  books  cost  c  dollars.  The  first  cost  d  cents  more 
than  the  second.    Find  the  cost  of  each. 

12.  A  and  B  have  k  dollars  in  all.  If  A  gives  h  dollars  to  B 
they  have  equal  sums.    How  many  dollars  had  each  at  first  ? 

13.  If  A  gives  h  dollars  to  B,  they  will  have  equal  sums.  If 
B  gives  k  dollars  to  A,  A  will  have  twice  as  much  as  B.  How 
many  dollars  has  each  ? 

14.  If  A  gives  $10  to  B,  B  will  have  h  dollars  more  than  A. 
But  if  B  gives  k  dollars  to  A,  A  will  have  three  times  as  much 
as  B.    How  many  dollars  has  each  ? 

15.  A  and  B  have  together  $40.  A  gives  h  dollars  to  B,  after 
which  B  gives  k  dollars  to  A.  Then  they  have  equal  sums 
How  many  dollars  bad  ^ach  ^t  first  ? 


LINEAE  SYSTEMS  235 

16.  A  gives  T  dollars  to  B  and  then  has  \  as  much  money  as 
B.  Then  B  gives  $8  to  A  and  has  left  -I  as  much  money  as  A. 
How  many  dollars  had  each  at  first  ? 

17.  A  part  of  |1000  is  invested  at  a%  and  the  remainder 
at  Z>%.  The  yearly  income  from  both  investments  is  c  dollars. 
How  many  dollars  are  there  in  each  investment  ? 

18.  A  portion  of  x  dollars  is  invested  at  5  %  and  the  remainder 
at  4%.  The  yearly  income  is  y  dollars.  How  many  dollars  are 
there  in  each  investment  ? 

19.  A  works  three  times  as  fast  as  B.  Together  they  can 
do  a  piece  of  work  in  c  days.  How  many  days  would  each 
require  alone  ? 

Hint.  Let  a  and  h  denote  the  number  of  days  required  by  A  and  B 
respectively  to  do  the  work  alone. 

Then  3  a  =  6,  and  -  +  -  =  -. 

a      h      c 

20.  A  works  h  times  as  fast  as  B.  Together  they  can  do  a 
piece  of  work  in  4  days.  How  many  days  would  each  require 
alone  ? 

21.  A  and  B  together  can  do  a  piece  of  work  in  d  days. 
A  can  do  -I  of  the  work  in  6  days.  How  many  days  does 
each  require  alone  ? 

22.  A  and  B  together  can  do  a  piece  of  work  in  5  days. 
A  can  do  -|  of  the  work  in  k  days.  How  many  days  does  each 
require  alone  ? 

23.  B  requires  twice  as  much  time  as  A  to  do  a  piece  of 
work  which  they  can  do  together  in  n  days.  How  many  days 
does  each  require  alone  ? 

24.  A  and  B  together  can  do  a  piece  of  work  in  jp  days. 
A  works  q^  times  as  fast  as  B.  How  many  days  does  each 
require  alone  ? 


236  FIRST  COURSE  IN  ALGEBRA 

100.  Linear  systems  in  three  unknowns.  The  method 
of  obtaining  the  set  of  roots  of  a  system  of  hnear  equa- 
tions in  three  unknowns  is  illustrated  in  the  following: 

EXAMPLE 

Ux-2y-{-z  =  9,  (1) 

Solve  the  system  J  3x  +  ij  +  2z  =  13,      .  (2) 

[2x  +  3i/-Sz=-2.  (3) 

Solution.  Eliminate  one  unknown,  say  z,  between  (1)  and  (2),  thus : 
(1)  .  2,  Sx-4:y  +  2z=lS.  (4) 

(2),  3  a:  +  ?/  +  2  2  =  13.  (5) 

(4)-(5),  5x-by=5,  (6) 

or  X  —  y  =  1.  *  (7) 

Now  eliminate  z  between  (1)  and  (3),  as  follows : 

(1) .  3,  12x-Qy  +  3z  =  27.  (8) 

(3),  2x-{-Sy-^z  =  -2.  (9) 

(8)  +  (9)  Ux-^i/  =  25.  (10) 

Solving  (7)  and  (10)  we  obtain         x  =  2,  y  =1, 

Substitute  2  for  x  and  1  for  y  in  (2), 

6  +  1  +  2  2  =  13.  (11) 

Solving  (11),  ^  =  3.  (12) 

Check.    Substituting  2  for  x,  1  for  y,  and  3  for  z  in  (1),  (2),  (3), 

8-2  +  3  =  9,         or         9  =  9. 

6  +  1  +  6  =  13,       or       13  =  13. 

4  +  3-9=- 2,     or     -2  =  -2. 

The  foregoing  example  illustrates  the 
Rule.  Decide  from  an  inspection  of  the  coefficients  which 
unknown  is  most  easily  eliminated. 

Using  any  two  equations^  eliminate  that  unknown. 
With  one  of  the  equations  just  used,  and  the  third  equation^ 
again  eliminate  the  same  unknown. 


LINEAR  SYSTEMS  237 

The  last  two  operations  give  two  equations  in  the  same  two 
unknowns.  Solve  these  two  equations  hy  the  rule  (^pp-  220-221), 

Substitute  the  two  values  found  in  the  simplest  of  the  origi- 
nal equations  and  solve  for  the  third  unknown. 

Check.  Substitute  the  values  found  in  each  of  the  original 
equations  and  simplify  results, 

EXERCISES 

Solve  for  the  unknowns  involved : 


x-\-y  +  z  =  ^, 

Sx-y-2z=-2, 

1. 

x-y  -z=-Z, 

8. 

6x  +  z  =  4^, 

X  -\-  y  —  z  =  5, 

Sy-4.Z=^11. 

X  +  y  -{-  z  =  S, 

x-\-y  =  l, 

2. 

2x-{-y  —  z  =  6, 

9. 

2/  +  ^  =  3, 

x-y-2z  =  0. 

z^x=^. 

3. 

X  -\-  y  -i-  z  =  0, 

Sx  +  2y-i-3z=-l, 

x-y-2z=-S. 

10. 

2p,-3p^  =  4., 
Sp,+p^  =  5, 
P,-2p^=2. 

SA-{-B-C  =  -S, 

11. 

6x-^3y-5z=^-S, 

4. 

A-AB-{-2C  =  9, 

4:X-\-6y  +  Sz  =  25. 

2^  +  35  +  3C  =  13. 

SA^  +  2A^  +  4.A^  =  9, 

Sx-2y-\-4,z  =  9, 

12. 

^1-^2-2^3=3^ 

5. 

2x  +  Sy-2z=-3, 

^1-3^, +  2^3  =2. 

5x-]-2y-\-Sz  =  6. 

Ill, 

6  A  -f  5  A:  +  4  m  =  9,  . 

x      y      z 

6. 

4.h-\-6k-i-5m  =  5, 

13. 

111       2 

1 \ =    o> 

5A  +  4A;  +  6m=16. 

x      y      z       3 
1       1   .   1       A 

x-^Sy-\-2z=lT, 

h  -  =  0. 

X      y      z 

7. 

y-Az=-5, 

Hint.  Solve  without  clearing  ( 

x-\-2y=-S. 

fractions. 

238  FIEST  COUESE  IN  ALGEBRA 

2      10_3^_  ?,§_§1 

p        q        r              '  X       z        ^ 

14.i  +  5  +  6^15,  15.5  +  ^  =  24, 

p       q       r  X       y 

i  +  ^-i=-^.  ^  +  ^  =  28. 

p      q      T          z  ^      y 


PROBLEMS 

1.  Eind  three  numbers  of  which  the  sum  of  the  first  and 
second  is  54,  the  second  and  third  Q>5,  and  the  first  and  third  59. 

2.  The  sum  of  three  numbers  is  70.  The  sum  and  the 
quotient  of  two  of  them  are  45  and  5  respectively.  Eind  the 
numbers. 

3.  The  perimeter  of  a  triangle  is  60  feet.  Two  of  its  sides 
are  equ^,  and  the  third  side  is  6  feet  longer  than  either  of  the 
first  two.    Eind  the  length  of  each  side. 

4.  The  sum  of  two  sides  of  a  triangle  is  53  feet  and  their 
difference  is  9  feet.  The  perimeter  of  the  triangle  is  72  feet. 
Eind  the  length  of  each  side. 

5.  The  sum  of  the  two  sides  of  a  triangle  which  meet  at 
one  vertex  is  41  feet,  at  another  vertex  46  feet,  and  at  the 
third  vertex  57  feet.    Eind  the  length  of  each  side. 

6.  The  sum  of  three  numbers  is  24.  The  quotient  of  two 
of  them  is  3,  and  the  sum  of  these  two  divided  by  the  third 
is  5.    Eind  the  numbers. 

Fact  from  Geometry.  The  sum  of  the  three  angles  of  any  tri- 
angle (each  angle  being  measured  in  degrees)  is  180  degrees. 

7.  Two  of  the  angles  of  a  triangle  are  equal,  and  their  sum 
is  equal  to  the  third.   Eind  the  number  of  degrees  in  each  angle. 

8.  Angle  ^  of  a  triangle  is  12  degrees  greater  than  angle  B, 
and  angle  B  is  21  degrees  greater  than  angle  C,  How  many 
degrees  are  there  in  each  ? 


LIISrEAR  SYSTEMS  239 

9.  The  sum  of  two  angles  of  a  triangle  is  30  degrees  more 
than  the  third,  and  the  third  is  15  times  the  difference  of  the 
first  two.    How  many  degrees  are  there  in  each  ? 

10.  A  and  B  together  can  do  a  piece  of  work  in  6  days, 
A  and  C  in  8  days,  and  B  and  C  in  12  days.  Eind  the  time 
required  by  each  alone  and  by  all  together. 

11.  Two  pumps  together  can  fill  a  tank  in  6  hours.  The  first 
of  these  and  a  third  together  can  fill  the  tank  in  8  hours.  All 
three  together  can  fill  the  tank  in  4  hours.  Eind  the  number 
of  hours  required  by  each  alone. 

12.  The  sum  of  the  digits  of  a  three-digit  number  is  19. 
The  units^  digit  exceeds  the  tens'  digit  by  3.  If  495  be  added 
to  the  number,  the  result  is  expressed  by  the  digits  in  reverse 
order.    Eind  the  number. 

13.  If  the  tens'  and  units'  digits  of  a  three-digit  T^lmber  be 
interchanged,  the  resulting  number  is  54  less  than  the  original 
number.  If  the  tens'  and  hundreds'  digits  be  interchanged, 
the  resulting  number  is  360  more  than  the  original  number. 
The  sum  of  the  digits  is  11.    Eind  the  original  number. 

Note.  Perhaps  the  student  wonders  whether  a  linear  equation  in 
three  unknowns  has  a  graphic  representation.  It  may  partially  satisfy 
his  curiosity  to  say  that  by  means  of  three  axes  at  right  angles  to  each 
other  such  a  representation,  though  beyond  the  scope  of  this  book, 
is  possible.  Further,  the  points  whose  x,  y,  and  z  values  satisfy  the 
equation  lie  in  a  flat  surface  called  a  plane.  Two  such  surfaces  may 
intersect  in  a  straight  line,  and  the  system  of  two  equations  which 
the  surfaces  represent  is  satisfied  by  the  x,  y,  and  z  values  of  any 
point  on  this  line.  If  three  such  surfaces  intersect  in  a  single  point, 
the  system  which  the  surfaces  represent  is  satisfied  by  the  a:,  y,  and  z 
values  of  this  point.  In  the  systems  of  equations  in  three  unknowns 
on  page  237  the  student  is  really  finding  the  coordinates  of  the  point 
of  intersection  of  three  planes. 

Since  space  has  but  three  dimensions,  this  method  of  representa- 
tion of  linear  equations  in  two  or  three  unknowns  cannot  be  extended 
to  equations  containing  four  or  more  unknowns. 


CHAPTER  XIX 

SQUARE  ROOT 

101.  Square  root  of  algebraic  expressions.  The  square 
root  of  f  +  2tu-\-u'  is   ±{t+  u). 

A  study  of  this  form  will  enable  us  to  extract  the  square 
root  of  any  polynomial.  Obviously  the  square  root  of  t'^ 
(the  first  term  of  the  trinomial)  is  t^  the  first  term  of  the 
root.  If  ^2  is  subtracted  from  the  trinomial,  the  remainder 
is  2tu-\-  u^.  The  next  term  of  the  root  (u)  can  be  found 
by  dividing  the  first  term  of  the  remainder  (2  tu)  by  2  ^ 
(twice  that  term  of  the  root  already  found). 

The  work  may  be  arranged  thus  : 

Trial  divisor,  2  t 


Complete  divisor,     2t  -{-  u 


2  m  +  w2 

2tu  +  u^  =  (2t-\-  u)u 


Therefore  the  required  roots  are  ±  (t  -{■  u). 

The  foregoing  process  is  easily  extended  to  extracting  the 
square  root  of  the  polynomial  4a?^— 20a;^+37a::^— 30a;  +  9, 
whose  square  root  contains  three  terms,  as  follows: 

(2x2)2=    Ax^ 


First  trial  divisor,  2  •  2^2=4:^2 
First  complete  divisor,  4^x^—5x 
Second  trial  divisor, 

2(2x^-6x)=4.x^-10x 
Second  complete  divisor,  4 a;2— 10x4-3 

240 


20x3+37x2 
20xH25x2=:(4x2-5x)(-5x) 


12x2-30x  +  9 

12x2-30x-f9  =  (4x2-~10x  +  3)8 


SQUAEE  EOOT  241 

Therefore  the  required  roots  are  dz  (2  a;^  —  5  a:  +  3). 

The  terra  2  x^  was  obtained  by  taking  the  square  root  of  4  a;*; 
the  second  term,  —  5  a;,  by  dividing  —  20  7^  by  the  first  trial  divisor, 
4  x^ ;  and  the  third  term,  3,  by  dividing  12  x^  by  4  x^,  the  first  term 
of  the  second  trial  divisor. 

The  method  just  illustrated  may  be  stated  in  the 

Rule,  Arrange  the  terms  of  the  polynomial  according  to 
descending  powers  of  some  letter  in  it. 

Extract  the  square  root  of  the  first  term.  Write  the  result 
(with  plus  sign  only)  as  the  first  term  of  the  root  and  sub- 
tract its  square  from  the  given  polynomial. 

Double  the  root  already  found  for  the  first  trial  divisor^ 
divide  the  first  term  of  the  remainder  by  it,  and  write  the 
quotient  as  the  second  term  of  the  root. 

Annex  the  quotient  just  found  to  the  trial  divisor,  making 
the  complete  divisor;  multiply  the  complete  divisor  by  the 
second  term  of  the  root  and  subtract  the  product  from  the 
last  remainder. 

If  terms  of  the  polynomial  still  remain,  double  the  root 
already  found  for  a  trial  divisor,  divide  the  first  term  of  the 
trial  divisor  into  the  first  term  of  the  remainder,  write  the 
quotient  as  the  next  term  of  the  root,  form  the  complete  divi- 
sor, and  proceed  as  before  until  the  process  ends,  or  until  the 
required  number  of  terms  of  the  root  have  been  found. 

Inclose  the  root  thus  found  in  a  parenthesis  preceded  by 
the  sign  ±, 

N'oTE.  The  process  of  extracting  the  square  root  of  numbers  was 
familiar  to  mathematicians  long  before  they  knew  how  to  find  the 
square  root  of  polynomials.  This  is  consistent  with  the  fact  that 
the  development  of  the  methods  of  performing  operations  on  literal 
number  symbols  generally  followed  and  grew  out  of  the  similar 
operations  on  numerals.  The  application  of  the  rules  for  extract- 
ing the  square  root  of  numbers  to  that  of  polynomials  is  generally 


242  FIRST  COUESE  IN  ALGEBKA 

ascribed  to  Recorde  (1510-1558),  who  was  the  author  of  the  earliest 
English  work  on  algebra  that  we  know.  This  book,  which  bears 
the  title  "The  Whetstone  of  Wit,"  gives  an  accurate  idea  of  the 
algebraic  knowledge  of  the  time,  and  had  a  very  wide  influence. 


EXERCISES 

Obtain  one  square  root  of : 

1.^2+6^  +  9.  4.  4aV+4a;*+a^ 

2.  x''-^10ax-\-25a\  5.  a'+ Sa^-i- 2  a^-^  2  a +  1. 

3.  x^-{-lQ-Sx\  6.  24:x^-32x-i-16  +  x^~-8x\ 

7.  x^  +  4  x^y  +  6  a^y  4-  4  xy^  +  y*. 

8.  21cHc*+20c-10c^+4. 

9.  7^^+9  71^+10  71^+25- 6  71^-30  71. 

10.  c^-12c  +  9c^+4  +  4c2-6cl 

11.  5  a^+  12  a^  16  -  23  a2+  4  a«  +  8  «^  -  22  a^ 

12.  c'-  4  c^a  +  6  cV-  4  ca^+  a\ 

13.  30xt/-^25y'-llxy-12xhj  +  4rx\ 

14.  -  36  a'^x  +  36  aV+  9  a^-  24  aV  +  16 x^+  48  ax«. 

15.  9c^-2a2^V+4a^Z>^c  +  a*Z/^-12a^cl 
16.2  w'xc^  _  4  ccc^  -  4  a'aj2  +  4  ic^  +  c^  +  a^'x^ 

5        25  a^      4 

19.  x^-Ax^+5x^-2x  +  h 


20.^^+4.3+4.^-^-^  +  1 

„    25m*      127 m^      „       ,   10m»  ,   9 
^^-l 18 2m  +  ^-  +  j. 


SQUAEE  EOOT  243 

102.  Square  root  of  arithmetical  numbers.  Since  1  =  1^ 
and  81  =  9^,  a  one-digit  or  a  two-digit  square  has  only  one 
digit  in  its  square  root. 

And  as  100=102,  and  9801  =  (99)2,  ^  three-digit  or  a 
four-digit  square  has  two  digits  in  its  square  root. 

Also  10,000=1002,  and  998,001  =  (999)2;  h^nce  a 
five-digit  or  a  six-digit  square  has  three  digits  in  its 
square   root. 

These  examples  illustrate  the  relation  between  the  num- 
ber of  digits  in  a  number  and  the  number  of  digits  in  its 
square  root.  They  also  suggest  a  method  of  obtaining  the 
first  digit  in  the  square  root  of  any  number. 

For  example,  take  the  four  numbers  78^43'56,  7^84^35, 
.98'01,  and  .03^27^4.  Beginning  at  the  decimal  point  in 
each,  point  off  periods  of  two  digits  each,  as  indicated. 
Any  period  incomplete  on  the  right,  as  in  .03^27.^4,  should 
be  completed  by  annexing  one  zero ;  thus,  .03^27^40.  Now 
the  first  digit  in  the  square  root  is  the  greatest  integer 
whose  square  is  less  than  or  equal  to  the  left-hand  period. 
This  is  true  whether  the  latter  contains  two  digits  or  one. 
Hence  the  first  digit  in  the  square  root  of  78'43'56  is  8, 
in  the  square  root  of  7'84'35  is  2,  in  the  square  root  of 
.98^01  is  9,  and  in  the  square  root  of  .03^27^40  is  1. 

Moreover,  the  number  of  digits  in  the  square  root  of  a 
perfect  square  is  equal  to  the  number  of  periods,  provided 
a  single  digit  remaining  on  the  left  is  called  a  period. 

Just  how  t  and  u  are  involved  in  the  square  of  (t-[-u)^ 
or  ^2  _j_  2  ^^^  4- 11^^  is  obvious  on  inspection,  because  the  parts 
t\  2tu^  and  u^  cannot  be  united  into  one  term.  In  the 
square  of  an  arithmetical  number,  however,  the  parts  are 
united.  Thus  (53)2  =  (50  +  3)2  =  2500  +  300  +  9  =  2809. 
Now  it  is  clear  how  50  and  3  are  involved  in  2500  -h  300  +  9, 
but  it  is  not  plain  from  2809  alone.   Pointing  off,  however, 


244 


FIEST  COURSE  UsT  ALGEBEA 


enables  us  to  discover  at  once  the  first  digit,  5,  which  is 
equivalent  to  5  tens,  or  50.  With  the  exception  of  pointing 
off,  the  method  of  extracting  the  square  root  of  an  arith- 
metical number  does  not  differ  greatly  from  the  method  of 
extracting  the  square  root  of  an  algebraic  expression.  In 
fact,  the  formula,  the  square  root  oit^+2  tu-\-u^  =  ±(t  +  u), 
can  be  used  to  explain  the  two  processes. 

If  t  denotes  the  tens  and  u  the  units,  t'^-}'2tu  +  u^  is 
closely  related  to  2500  +  300  +  9,  t^  being  2500,  or  (50)2 . 
u^  being  9,  or  3^ ;  and  2  tu  being  2  .  50  .  3.  Therefore  the 
process  of  extracting  the  square  root  of  2809  may  be  based 
on  these  relations  and  the  work  arranged  as  follows: 

2809(50  +  3 
^2  =  (50)2  2500 

2^  =  2. 50=100  r309 
2^+^^=100  +  3  309=(100  +  3)3=(2^  +  ^)i^  =  2^t^  +  i/2 


Therefore  ±  53  are  the  two  square  roots  of  2809. 
If  the  number  has  three  digits  in  its  square  root,  the 
work  and  explanations  may  be  arranged  thus: 


^2  =(100)2 
First  trial  divisor, 

2^=2.  100  =  200 
First  complete  divisor, 
2^  +  2^  =  200  +  30  =  230 
Second  trial  divisor, 

2^=2.  130  =  260 
Second  complete  divisor, 
2^  +  1^=260  +  2  =  262 


174^24|100  +  30  +  2 

1  00  00  =  10  tens  squared 


74  24 


69  00  =  (2  .  10  tens  +  30  units)  30 


5  24 


5  24  =  (2.13  tens  +  2  units)  2 


Therefore  ±  132  are  the  square  roots  of  17,424. 


SQUAEE  EOOT  245 

When  the  method  and  reasons  for  the  process  have 
become  familiar,  the  work  may  be  shortened  by  omitting 
the  explanations  and  unnecessary  zeros  as  follows: 


28'09|53 

174'24  132 

25 

1 

309 

23 

74 

309 

69 

262 

524 

524 

The  method  of  the  two  preceding  solutions  is  the  one 
commonly  used  for  extracting  the  positive  square  root  of 
a  number.    For  it  we  have  the 

Rule.  Begin  at  the  decimal  point  and  point  off  as  many 
periods  of  two  digits  each  as  possible :  to  the  left  if  the 
number  is  an  integer ;  to  the  right  if  it  is  a  decimal ;  to  both 
the  left  and  the  right  if  the  number  is  part  integral  and 
part  decimal. 

Find  the  greatest  integer  whose  square  is  equal  to  or  less 
than  the  left-hand  period^  and  write  this  integer  for  the  first 
digit  of  the  root 

Square  the  first  digit  of  the  root^  subtract  its  square 
from  the  first  period^  and  annex  the  second  period  to  the 
remainder. 

Double  the  part  of  the  root  already  found  for  a  trial  divisor^ 
divide  it  into  the  remainder  (omitting  from  the  latter  the 
right-hand  digif),  and  write  the  integral  part  of  the  quotient 
as  the  next  digit  of  the  root. 

Annex  the  root  digit  just  found  to  the  trial  divisor  to 
make  the  complete  divisor^  multiply  the  complete  divisor  by 
this  root  digits  subtract  the  result  from  the  dividend,  and 
annex  to  the  remainder  the  next  period  for  a  new  dividend. 


246  FIRST  COUESE  IN  ALGEBRA 

Double  the  part  of  the  root  already  found  for  a  new  trial 
divisor  and  proceed  as  before  until  the  desired  number  of 
digits  of  the  root  have  been  found. 

After  extracting  the  square  root  of  a  number  involving 
decimals,  point  off  one  decimal  place  in  the  root  for  every 
decimal  period  in  the  number. 

Check,  If  the  root  is  exact,  square  it.  The  result  should 
be  the  original  number.  If  the  root  is  inexact,  square  it 
and  add  to  this  result  the  remainder.  The  sum  should  be 
the  original  number. 

Sometimes  in  using  a  trial  divisor  we  obtain  too  great  a  quotient 
for  the  next  digit  of  the  root.   This  happens  in  obtaining  the  second 
digit  of  the  square  root  of  32,301,  where  2  into  22 
gives  11.    Obviously  10  and  11  are  both  impossible.  ^  l_ 

If  9  is  tried  we  get  9  •  29,  or  261,  which  is  greater 


O    OOQ 

than  223.    Similarly,  8  is  too  great.    But  7-  27^:189,         ' 

which  is  less  than  223.    Therefore  7  is  the  second  digit  of  the  root. 

Occasionally  the  trial  divisor  gives  a  quotient  less  than  1.    This 
indicates  that  the  required  root  digit  is  0,  which  should  be  written 
in  the  root  and  the  work  continued  as  usual.    An 
instance  of   this  kind  occurs  in  finding  the  second        9^42.49 1 30 
digit  in  the  square  root  of  9^42.49.    The  quotient  of        9 
4  ^  6  is  §,  which  is  not  an  integer.     Therefore  the    6 1  42 
second  digit  of  the  root  is  0.    Then  the  next  period, 
49,  should  be  brought  down.    The  new  trial  divisor  will  be  60, 
which  will  give  7  as  the  third  digit  of  the  root.     The  work  can 
easily  be  completed,  giving  30.7  as  the  square 
root.  •   3.00^00^00 1 1.732 

An  attempt  to  extract  the  square  root  of  3  1_ 

by  annexing  decimal  periods  of  zeros  and  ap- 
plying the  rule  becomes  a  never-ending  process. 

The  number  3  has  no  exact  square  root,  and 
no  matter  how  far  the  work  is  carried,  there  is 
no  final  digit.  As  the  work  stands,  we  know  that 
the  square  root  of  3  lies  between  1,732  and  1.733. 


27  ^ 

>00 

1 

L89 

343~ 

1100 

1029 

346^ 

I     7100 

6924 

SQUARE  EOOT  247 

EXERCISES 

Obtain  the  positive  square  root,  to  three  decimal  places,  of: 

1.  4489.  4.  6241.  7.  24649.  10.  165680. 

2.  5184.  5.  9216.  8.  43436.  11.  223729. 

3.  5329.  6.  16129.  9.  53361.  12.  328329. 

Fact  from.  Geometry.  In  the  adjacent  right  triangle, 
a^  -I-  ^2  _  ^2 .  ^^  sides  a  and  &,  which  form  the  right  angle, 
are  called  the  legs ;  and  c,  the  side 
opposite  the  right  angle,  is  called 
the  hypotenuse. 

If  leg  a  is  8  and  leg  h  is  15, 
then  substituting  in  a^  +  ly^  =  c^ 
gives  64  +  225  =  c2.  Whence  289 
=  c^  and  c=±ll. 

Since  —  17  is  not  a  practical  answer,  it  is  rejected. 

In  Exercises  13-16  find  the  hypotenuse  and  the  area  of  a 
right  triangle  whose  legs  are  : 

13.  63  and  16.  15.  104  and  153. 

14.  48  and  B^.  16.  645  and  812. 

In  Exercises  17-19  find  the  other  leg  and  the  area  of  a  right 
triangle  in  which  the  hypotenuse  and  one  leg  are  respectively  : 

17.  109  and  91.         18.  2.57  and  .32.         19.  2.05  and  1.87. 

Extract  the  square  root  in  Exercises  20-23  inclusive  to  four 
decimals,  and  in  Exercises  24-28  inclusive  to  three  decimals. 

A  common  fraction  or  the  fractional  part  of  a  mixed  number  should 
be  reduced  to  a  decimal  before  extracting  the  square  root,  unless  the 
root  is  seen  to  be  exact. 


20.  6.4271. 

23. 

.00321. 

26.  2f. 

21.  884.3. 

24. 

3. 

27.  2f 

22.  .0869. 

RE 

25. 

6. 

28.  -ij 

248 


FIKST  COUESE  IN  ALGEBEA 


(In  the  following  find  all  inexact  answers  to  two  decimals.) 
In  rectangle  ABCD  line  DB  is 
called  a  diagonal. 

29.  Find  the  diagonal  of  a  rec- 
tangle whose  adjacent  sides  are 
28  feet  and  195  feet  respectively. 

30.  One  diagonal  of  a  rectangle 
is  409  and  one  side  is  391.    Find  the  other  side  and  the  area. 

31.  One  diagonal  of  a  rectangle  is  533  and  one  side  is  92. 
Find  the  perimeter  of  the  rectangle. 

32.  A  rectangle  is  7  yards  longer  than  it  is  wide.    Its  per' 
imeter  is  138  feet.    Find  one  diagonal. 

33.  One  diagonal  of  a  square  is  74  meters.    Find  the  side. 

34.  The  side  of  a  square  is  52  inches.    Find  one  diagonal. 

35.  One  leg  of  a  right  triangle  is  10.    The  hypotenuse  is 
twice  the  other  leg.    Find  the  hypotenuse  and  the  other  leg. 

36.  The  hypotenuse  of  a  right  triangle  is  three  times  one 
leg  and  the  other  leg  is  16.    Find  the  sides. 

37.  A  rectangle  is  2.4  times  as  long  as  it  is  wide.    One 
diagonal  is  52.    Find  the  length  and  the  width. 

38.  The  width  of  a  rectangle  is  25%  less  than  the  length. 
The  diagonal  is  100.    Find  the  area. 

39.  The  length  of  a  rectangle  is  10.    The  diagonal  is  twice 
the  shorter  side.    Find  the  width. 

Fact  from  Geometry.  A  line  drawn 
from  one  vertex  of  an  equilateral  tri- 
angle to  the  middle  point  of  the  oppo- 
site side  is  perpendicular  to  that  side. 

Then,  in  the  equilateral  triangle 
ABC^  if  D  is  the  middle  point  of  ^C, 
BD  is  the  altitude ;  and 

bd'=  ab'-  ad'=  ab'-  (^Y- 


SQUAKE  EOOT  249 

40.  If  BC  in  the  foregoing  triangle  is  6,  find  BD  and  the 
area  of  the  triangle. 

41.  If  AC  in  the  foregoing  triangle  is  8,  find  J5i)  and  the 
area  of  the  triangle. 

42.  If  BD  in  the  foregoing  triangle  is  14,  find  AB  and  the 
area  of  the  triangle. 

43.  The  perimeter  of  an  equilateral  triangle  is  45.  Find  the 
altitude. 

44.  The  altitude  of  an  equilateral  triangle  is  25  centimeters. 
Find  one  side. 

Note.  A  method  of  extracting  the  square  root  of  numbers  not 
unlike  that  in  use  to-day  was  employed  by  the  Greek,  Theon,  about 
A.D.  350.  In  the  Middle  Ages  square  roots  were  extracted  with  a  fair 
degree  of  accuracy  by  using  the  formulas  of  approximation : 

(1)   Va^+x  =  a  +  -^.  (2)  Va^+x  =  a  +  —^. 

2a  ^  ^  2a  +  l 

The  true  value  of  the  square  root  of  the  number  was  proved  to  be 
between  the  results  obtained  by  these  expressions.  Thus,  if  V65  was 
desired,  it  was  noticed  that  65  =  64  +  1,  and  from  (1) 

V65  =  V64  +  1  =  V82  +  1  =  8  +  -^  =  8t-V, 

2  •  8 
while  from  (2) 

V65  =  V64+1  =  V82+l=  8  + =  8tV 

2.8  +  1 

Thus  the  true  value  of  V65  is  between  these  two  numbers.  Thi* 
method  was  known  to  the  Arabs. 

It  should  be  kept  in  mind  that  the  use  of  decimal  fractions  and 
of  the  decimal  point  was  not  common  until  the  eighteenth  century. 
Consequently  the  complete  development  of  the  method  of  extract' 
ing  the  square  root  given  in  the  text  is  comparatively  recent. 


CHAPTER  XX 

RADICALS 

103.  Rational  numbers.  The  quotient  of  two  integers 
is  called  a  rational  number. 

Any  integer  is  a  rational  number,  since  it  may  be  con- 
sidered as  the  quotient  of  itself  and  1. 

Thus  5,  —  8,  I,  —  I,  and  4.693  are  rational  numbers. 

'    104.  Radical.    A  radical  is  an  indicated  root  of  an  alge- 
braic or  arithmetical  expression. 

Thus  Vi,  V3,  Vay  and  Vx^  —  5  a;  +  6  are  radicals. 

If  a  number  under  a  radical  sign  is  such  that  the 
root  cannot  be  taken  exactly,  the  radical  represents  an 
irrational  number.  G 

Thus  V2,  ^2,  Vs,  ^5,  are  irrational  num- 
bers, since  the  indicated  roots  of  2,  of  3,  and 
of  5  will  never  come  out  even  however  far 
the  process  of  extracting  the  root  is  carried. 

Though  no  irrational  numbers  can  be  ex- 
pressed exactly  in  decimals,  we  can  represent 
a  few  of  them  by  the  lengths  of  lines.    Thus 
in  the  right  triangle  ABC,  if  AB  =  AC=1  inch,  BC=V2  inches. 
If  AB  were  2  inches  and  AC  were  1  inch,  BC  would  be  Vs  inches. 

There  are  other  types  of  irrational  numbers  which  cannot 
be  expressed  in  terms  of  radicals,  but  their  consideration 
is  too  complicated  for  this  text. 

If  a  negative  number  occurs  under  a  square-root  sign, 
the  radical  represents  an  imaginary  number. 

250 


RADICALS  251 

Thus  V—  2,  V  —  8,  and  V—  4  are  imaginary  numbers. 

If  the  student  pursues  the  study  of  algebra  he  will  learn  that 
imaginary  numbers  are  required  to  express  completely  the  cube 
and  higher  roots  of  any  positive  or  negative  number. 

For  example,  he  will  learn  that  the  number  27  has  two  other  cube 
roots  besides  the  number  3. 

105.  Index.  The  small  figure  like  the  4  in  "v^  is  called 
the  index  of  the  radical. 

The  index  determines  the  order  of  the  radical  and  indi- 
cates the  root  to  be  extracted. 

In  5  Vl,  3  is  the  index,  and  the  radical  is  of  the  third  order. 

106.  Radicand.  The  radicand  is  the  number,  or  expres- 
sion, under  the  radical  sign. 

In  Vy  and  Vox,  7  and  ax  are  the  radicands. 

107.  Principal  root.  For  a  given  index  the  principal  roet 
of  a  number  is  its  one  root^  if  it  has  but  ope ;  or  its  positive 
root,  if  it  has  two  roots. 

The  principal  root  of  ^-27  is  -  3  ;   that  of  ^^16  is  +  2,  not  -  2. 

108.  Fractional  exponents.  Radical  expressions  may  be 
written  in  two  ways,  with  radical  signs  or  with  fractional 
exponents.  The  relation  between  the  two  will  now  be  ex- 
plained. To  do  this  it  is  necessary  to  extend  the  meaning 
of  the  term  exponent,  which  as  defined  on  page  9  applied 
to  positive  integral  exponents  only.  We  shall  assume  that 
the  laws  which  govern  the  use  of  integral  exponents  hold 
for  fractional  exponents  also. 

The  fact  that  x^  -  a^  =  cifi  illustrates  the  more  general 
law  x^  *  x^  =  x^-^\  where  a  and  b  represent  either  integeie 
or  fractions. 

Accordingly  x^  >  x'^  =  x'^'^'^  =  x^  or  x.  Since  x^  multiplied 
by  itself  gives  x,  x^  must  he  another  way  of  writing  the  square 
root  of  X. 


252  FIRST  COUESE  IN  ALGEBEA 

Hence  V^  may  be  written  xk 


Then     4i=V4:  =  2,     and     (25a^)i  =  y/2^a^  =  5  a, 
Further,  x'^  »  xi  -  x^  =  x^  =  x. 

And  since  x^  is  one  of  the  three  equal  numbers  whose 
product  is  a:,  x^  is  another  way  of  writing  the  cube  root  of  x. 
Therefore  vi  may  be  written  a;^. 


This  means 

;  that 

8i=^8  =  2. 

Similarly, 

■\/x  =  a;i. 

In  general 

terms, 

nr         i 

Now 

x^  =  x^  .  x^  .  x^  =^{x^J  =  {^'x)\ 

and 

^f  ^^•i^^^y.^V^. 

Hence 

x^={^x)^,  or  V^. 

In  general 

terms. 

x-  =  ^^. 

Thus  x^  means  the  nth  root  of  x  to  the  ath  power. 

The  student  should  fix  in  mind  that  the  denominator 
of  the  fractional  exponent  is  the  index  of  the  root,  and 
the  numerator  the  power  to  which  the  radicand  is  raised. 
Moreover,  whether  one  extracts  the  root  first  and  then 
raises  the  result  to  the  power,  or  vice  versa,  depends  wholly 
on  convenience. 


ORAL  EXERCISES 

Read  in  radical  form : 

1.  x^,                         6.  5  a*. 

11. 

^a^ihcf 

2.  xK                         1.  (pa)i. 

12. 

5U 

3.  (cd)i.                    8.  3  raj*. 

4.  (2  a;)*                     9-  hh^. 

13. 

1  i 
1    c 

6.  2x^.                    10.  ls^(t^\- 

w)k 

14. 

2x^y^. 

EADICALS  253 

Pind  the  numerical  values  of : 

15.  25*.  20.  125*  25.  (J^)i. 

16.  27l  21.  (-  8)3.  26.  (^L)i 

17.  16*.  22.  32i  27.  (l\)l 

18.  4l  23.  8ll  28.  25*.  4^ . 

19.  64*.  24.  (-  216)1  29.  4*  •  (|)l 

30.  What  is  the  principal  square  root  of  4  ?  of  25  ?  of  36  ? 
the  principal  cube  root  of  +  8  ?  of  -  8  ?  of  -  27  ?  of  -f  27  ? 
the  principal  fifth  root  of  32  ?  of  -  32  ?  of  243  ?  of  -  243  ? 

31.  What  is  an  index  ?  a  radicand  ?  Illustrate. 

EXERCISES-. 

Write  with  fractional  exponents  and  simplify  results : 

8.  2-y/27\  15.  3V?.  -v^. 

9.  3  Vs^\  16.  -V^  .  -V^. 

10.  4  ^27  ax\  17.  V^.  V^. 

11.  2^^^:  18.  (-32)*.  ■v^-64. 

12.  4-v/T6^.  19.  36'  -Vi. 

13.  12x2-v^^.  20.  9*  .  -v^. 

14.  cV(^l  21.  (^)*.  -v^8^. 

109.  Simplification  of  radicals.    The  form  of  a  radical 

expression  may  be  changed  without  altering  its  numerical 

value.    Such  changes  are  necessary  for  many  reasons.    For 

example,  the  numerical  value  of   a  radical  expression  is 

most  easily  obtained  from  its  simplest  form.     It  will  be 

1       V2 
made  clear  later  that  -—  =  -—.    Granting  that  the  two 

fractions  are  really  equal,  one  can  see  by  inspection  that 
the  value  to  several  decimals  can  be  computed  more  easily 
from  the  second  fraction  than  from  the  first. 


1. 

V^. 

2. 

■\/ax\ 

3. 

3V2if^ 

4. 

V9;r. 

5. 

5  Vl6  ax-. 

6. 

^a\ 

7. 

^ax^. 

254  FIEST  COUESE  IN  ALGEBEA 

EXAMPLES 

Study  the  following  changes  of  form  : 

1.   V36  =  V479  =  Vi.  V9  =  2.  3  =  6. 
■     2.  Similarly,  Vs  =  V4T2  =  Vi  .  V2  =  2  V2. 

3.  More  generally,  Va^  =  Va^  ■\/ ab  =  a  -^/ab, 

4.  Also  -v^  =  -\/8T3  =  -v^8a/3  =  2^3. 

5.  More  generally,  "y/a^  =  V a^  V ^  =  a  V^. 

6.  Finally,  "v^  =Vi^^  =  aVb. 

The  six  preceding  examples  illustrate  the  correctness  of 
the  following  rule  for  simplifying  a  radical  involving  the 
square  root  of  an  integer  or  an  integral  expression. 

Rule.  Separate  the  radicand  into  two  factors  one  of  which 
is  the  greatest  perfect  square  which  it  contains.  Then  take 
the  square  root  of  this  factor  and  write  it  as  the  coefficient 
of  a  radical  of  which  the  other  factor  is  the  radicand. 

If  the  radical  already  has  a  coefficient  other  than  the 
number  i,  multiply  the  result  obtained  above  by  this  coefficient, 

A  similar  rule  holds  for  radicals  involving  the  cilbe  and 
higher  roots. 

Thus  </lQ  =  V^T^  =  2V2, 


and  V96  =  V32  •  3  =  2  Vs. 

Note.  Although  the  Arabs  were  by  no  means  able  to  state  all 
the  rules  explained  in  this  chapter,  it  is  interesting  to  note  that  they 
did  recognize  the  truth  of  a  few  of  them.  For  instance,  a  writer 
about  A.D.  830  gives,  in  his  own  notation,  of  course,  the  facts  con- 
tained in  the  formulas  Va^  =  a V^,  and  Va5  =  va  vh. 


RADICALS 


255 


Simplify : 

1.  Vs. 

2.  Vl2. 

3.  Vl8. 

4.  V20. 

5.  V45. 

6.  V5O. 

7.  V75. 

8.  V63. 

9.  V98. 

10.  2V72. 

11.  3V8O. 

12.  5V128. 

13.  Vl47. 

14.  3Vi62. 


EXERCISES 


16.  ^ 

17.  A 

18.  ^ 

19.  ^ 

20.  ^ 

21.  2 


'192. 
^243. 
^245. 

^363. 

V72O. 


22.  V1250. 

23.  VI6. 

24.  V24. 

25.  2^40. 

26.  3^56. 

27.  4^72. 

28.  5^96. 

29.  ^v^. 
15.  IOVI75.    30.  -y/si. 


57.  Vs  +  4  V2. 
Solution.    Vg  4.  4  V2  = 

V4(2  + V2)  = 
2  V2  +  V2. 

58.  \/4  -  4  V3. 

59.  Vis  4-  9  V3. 


31.   V135. 
3 


32.  3V189. 

33.  7^128. 

3/ 


44.  V 

45.  2 


46.  3  VS  a^x. 


34 
35 


^92. 

/250. 


47.  5a?V4aV. 

48.  V^. 


36.   SV375.       Hint. 


37.  ^448. 
3 


38.  2V625. 

39.  Va\ 

Hint.   Vo^  = 


-^a^ '  a, 

49.  -v^. 

50.  2^Vx^. 

51.  -W?, 


40. 
41. 

vi; 

Vs^ 
Vs^ 

60. 
61. 
62. 
63. 
64. 

52. 
53. 

?,         54. 
x\,       55. 
?.         56. 

V27cc^ 
3aj-v^4a8x. 
5  X  V4  a^aj^. 

42. 

■</27a'x. 

43. 

■\/56a^x. 

Vioo- 

\/9V2- 

25  V5. 

-27. 

^V3. 

rWB. 

-  251 

65.  VaV  +  o^Vx. 


The  foregoing  exercises  are  easier  to  simplify  than 
radicals  whose  radicands  are  fractions  or  fractional  expres- 
sions. The  latter  arise  frequently  in  practice,  especially 
in  certain  parts  of  geometry. 


256  FIRST  COUESE  IN  ALGEBRA 

EXAMPLES 

Study  the  following  simplification  of  fractional  radicands: 

1.  V|  =  V|=vT^  =  ViV6  =  jV6. 

2.  6V|  =  6V|  =  6VlT3  =  6.lV3  =  2V3. 

These  examples  illustrate  the  following  rule  for  sim- 
plifying a  square  root  ^hich  has  a  fractional  radicand. 

Rule.  Multiply  the  numerator  and  the  denominator  of  the 
radicand  hy  the  least  whole  number  or  simplest  expression 
which  will  make  the  resulting  denominator  a  perfect  square. 

Then  separate  the  radicand  into  two  factors^  one  of  which 
is  a  fraction  and  at  the  same  time  the  greatest  perfect  square 
which  the  radicand  contains. 

Take  the  square  root  of  this  factor  and  write  it  as  the 
coefficient  of  the  radical  of  which  the  other  factor  is  the 
radicand.  If  the  original  radical  has  a  coefficient^  multiply 
the  result  as  obtained  above  by  this  cofficient. 

A  similar  rule  holds  for  simplifying  a  cube  root  and 
radicals  of  higher  orders  which  have  fractional  radicands. 

EXERCISES 

Simplify  the  following : 

1.  V|.  6.  Vf.  11.  Vf.  16.  V^. 

2.  Vf.  7.  V|.  12.  Vf.  17.  SV^. 

3.  V^.  8.  Vf,  13.  sVf.  18.  2V^. 

4.  V|.  9.  V^.  14.  2V|.  19.  3aV^. 

5.  \/|.  10.  V|.  15.  Vf.  20.  3(^VS- 


RADICALS  257 


21.  Vl-ay.         24.  V25-(|/.  3^,       S^_ 

VI3T=  26.V81-(|)^  3,.^ 

V|,  etc.  27.  Vl21-(-y-)^. 


22.  Vi-a)^ 

23.  V9-(|)''. 
34 


28.  Vl69-(J^^)^        32.  V^M^l 


29.  \t-  33 


■  ^J--(f)' 


2  ■ 


•   >i^'-(fj-  38.  ^2E^V2 

36.  ^|f+^-  40.  V16  +  8V2. 


41.  V8I  +  3V243. 


The  need  for  simplifying  radicals  presents  itself  in  vari- 
ous problems,  as,  for  example,  in  simple  geometrical  work 
on  right  triangles. 

PROBLEMS 

(Obtain  answers  in  simplest  radical  form.) 

1.  One  leg  of  a  right  triangle  is  8  and  the  other  is  10.  Find 
the  hypotenuse. 

Solution.    X  =  V82  +  102  =  ■y,^]M  =  V4  •  41  =  2  ViT. 

2.  The  hypotenuse  of  a  right  triangle  is  8  and  one  leg  is  4. 
Find  the  other  leg  and  the  area. 

3.  The  hypotenuse  of  a  right  triangle  is  R  and  one  leg  is  —• 
Find  the  other  leg  and  the  area. 

4.  Find  the  diagonal  of  a  square  whose  side  is  12. 


258 


FIEST  COUKSE  IN  ALGEBRA 


5.  Find  the  sides  of  a  square  whose  diagonal  is  12. 

6.  Find  the  sides  of  a  square  whose  diagonal  is  2  R. 

Problems  involving  the  following  classes  of  triangles  are 
of  frequent  occurrence  in  practical  work  and  often  require 
the  use  of  radicals  : 

(a)  An   isosceles   right  triangle ;    that  is,  a 
right  triangle  with  two  equal  sides.  ^ 
.    As  .  indicated  in  the  figure,  if  each  leg  is  aJ? 
the  two  acute  angles  are  45°  each. 

(b)  A  right  triangle  with  one  angle  30°  or  60°. 

As  indicated  in  the  figure,  if  one  acute  angle  is  30°  the 
other  is  60°,  and  vice  versa.    More  important 
still,  the  hypotenuse  is  always  twice  the  shorter 
leg, 

(p)  An  equilateral  triangle. 

As  indicated  in  the  figure,  when  the  alti- 
tude is  drawn,  it  divides  the  base  into  two 
equal  parts,  and  in  each  of  the  right  triangles 
formed  the  same  relations  exist  as  in  (b),  above. 

7.  If  each  leg  of  an  isosceles  right  triangle  is  6,  find  the 
hypotenuse. 

Hint.   x2  =  62  +  62. 

aj  =  V36  +  36,  etc. 

8.  Find  the  hypotenuse  of  an  isosceles  right 
triangle  if  one  leg  is  8 ;  if  one  leg  is  13. 

9.  The  hypotenuse  of  an  isosceles  right  triangle  is  10. 
Find  the  legs. 

Hint.   x2  +  x2  =  100.   x2  =  50,  etc. 

10.  The  hypotenuse  of  an  isosceles  right  triangle  is  13. 
Find  the  legs. 

11.  One  angle  of  a  right  triangle  is  30°.  The  hypotenuse  is 
20.    Find  the  other  two  sides. 

Hint.   See  (6),  above. 


EADICALS  259 

12.  One  angle  of  a  right  triangle  is  60°.    The  hypotenuse  is 
12.    Find  the  other  two  sides. 

13.  One  angle  of  a  right  triangle  is  30°.    The  leg  opposite 
it  is  10.    Find  the  hypotenuse  and  the  other  leg. 

14.  One  angle  of  a  right  triangle  is  60°  and  the  adjacent  leg 
is  12.    Find  the  other  two  sides. 

15.  The   side   of  an  equilateral  triangle  is  12.     Find  the 
altitude  and  the  area. 

Hint.    See  (c)  and  (6),  page  258. 

16.  The  side  of  an  equilateral  triangle  is  S.    Find  the  alti- 
tude and  the  area. 

17.  The  altitude  of  an  equilateral  triangle  is  10.    Find  the 
side  and  the  area. 

18.  The  legs  of  a  right  triangle  are  equal.  Its  hypotenuse  is 
20.    Find  the  legs  of  the  triangle. 

19.  The  legs  of  a  right  triangle  are  equal  and  its  area  is  50. 
Find  the  hypotenuse. 

20.  The  legs  of  a  right  triangle  are  —  and  -^-    Find  the 
hypotenuse. 

21.  One  leg  of  a  right  triangle  is  -^-    The  hypotenuse  is  R. 
Find  the  other  leg. 

Find  X  and  y  in  terms  of  it  in  the  following : 


22.    a;  X  23. 


X 

p 

24.   The   legs    of   a  right   triangle   are  R  and   —(a—T), 

Find  the  hypotenuse. 

110.  Addition  and  subtraction  of  radicals.  Radicals  of 
the  same  order  which  are  in  their  simplest  form  and  have 
like  radicands  are  really  similar  terms.    They  can  be  added 


260  FIEST  COUESE  IN  ALGEBRA 

or  subtracted  and  the  result  expressed  by  one  term  accord- 
ing to  the  rule  on  page  33. 

Thus  3V2 -7V2  +  9  V2  =  5V2. 

Similarly,  7 Vj  -  |V2  =  ^  V2  -  fV2  =  2  V2. 

The  last  example  illustrates  the  necessity  of  acquiring 
the  ability  to  simplify  radicals  before  attempting  to  carry 
out  the  fundamental  operations  of  addition  and  subtraction. 

If  the  radicands  are  unlike  and  cannot  be  simplified 
further,  the  radicals  are  really  dissimilar  terms,  and  ad- 
dition or  subtraction  can  only  be  indicated.    (See  page  34.) 

Thus  V2,  V 3,  and  V5  are  three  dissimilar  radicals,  and  the  ad- 
dition of  the  three  can  only  be  indicated  thus  :  V2  +  Vs  +  Vo. 

EXERCISES 

Simplify  and  collect : 


1.  V27  +  V12. 

4. 

V28  + V63. 

2.   V45-V2O. 

5. 

7VI8- V98. 

3.  2V2OO-3V8, 

6. 

V76  -  V27  +  2  Vis. 

7.  V2  +  VI.          ^^ 

8.  V3  -  Vl. 

9.  f  +  V|. 

io.|  +  Vf.         ^'• 

R   ,      R 

37? 

2       > 

)R' 
2 

97?'' 

2 

13.  R  -  "vj- 

14.  5VI- 

15.  sVf- 

iR^ 
4 

fV2. 
V72. 

16.  V|  +  2^/|-J 

'Vf. 

19. 

•  V^+2V¥- 

-v^^. 

17-  Vl-ViV  +  2 

Vio. 

20. 

,  ^56  + 2^189, 

18.  V^-Vl20- 

•2V|. 

21. 

,  2  VS20  -  V50. 

22.   -v^-V20. 

Hint.  </2^  =  V^  =  ^i  = 

5i  =  A 

etc. 

-ilADICALS  .        261 

23.   ■V9  +  V12.  24.   -v^  + 5-^/162. 

25.  2xV5^-3Vl6^'-\-'-\/4^\ 

26.  -^ST^  +  X  VS75  x'  -  ^-v^ie^. 

27.  "^ o^hc  —  a^Tohc -{- ac-yA  —  • 

28.  ^5  4^4-^^-2-v^. 

29.  Va^  +  4  a^  f  4  (X  —  Va^ ^sTc^, 


30.   V3  x2  -  18  03  +  27  -  V27  (ic2  +  2  X  +  1). 

Note.  Though  methods  of  classifying  irrational  expressions  are 
found  in  the  work  of  Euclid,  the  Hindus  and  the  Arabs  were  the 
first  to  develop  this  part  of  algebra  in  anything  like  the  form  used 
to-day. 

111.  Multiplication  of  radicals.  Monomial  radicals  of 
the  same  order  are  multiplied  as  follows: 


EXAMPLES 

1.  Multiply  3  VS  by  2  V5. 

Solution.    3  Vs  .  2  V5  =  6  Vio  =  6-2  VlO  =  12  VlO. 

2.  Multiply  5  V^^  by  ^2  a V. 

Solution.  5  \/4  ax^  •  V%aV  ^  5  \/8  a^x^  =  10  ax  </x. 

The  method  just  illustrated  of  multiplying  monomial 
radicals  of  the  same  order  may  be  stated  in  the 

Rule.  Take  the  product  of  the  coefficients  of  the  radicals 
for  the  coefficient  of  the  radical  in  the  result. 

Multiply  together  the  radicands  and  write  the  product 
under  the  common  radical  sign. 

Reduce  the  result  to  its  simplest  form. 

The  preceding  rule  does  not  hold  for  radicals  of  different  orders. 


262 


FIEST  COUKSE  IN  ALGEBEA 


EXERCISES 

Perform  the  indicated  operation  and  simplify  results : 
1.  V5  .  Ve.  7.  5*  .  20*.  13.  R^'  R  Vs 

8.  18*  .  8^. 

9. 

10.  Vf.V-f. 

11.  vn .  Vt\ 

12.  a^  '  (bc)^. 


2.  Vt  .  V7. 

3.  2  Vs.  3V2. 

4.  3V2.5V2. 

5.  V2.V8. 

6.  V3  .  V27. 

17.  2V^.  7Va¥. 

18.  V75^.(45a)l 

19.  V2^  .  V4 1;  .  V6  uv. 

20.  5  V3m  .  5  VSm. 

21.  (3V3^)'. 

22.  V16  .  -v^. 

23.  ^4.-^12. 

24.  (100)*  .  (30)*. 


M-V75. 


14.(^|_V2 

^V2./^V3, 
4^1 


15 


2 
16.  2V^ 

25.  ^8.  V32. 

26.  5  -v^  .  3  Vl6^. 


27.  Wr  -  V2)'. 

28.  (|V23V5j 

m^        (4^ 

29.  A^-.  aJ 

30.  J^. 

X  a 


rind  the  radical  expression  having  the  coefficient  1  equiva- 
lent to  each  of  the  following : 
31.  5V2. 
Solution.    5  V2  =  V25V2  =  V50. 

34.  8\/j,         36.  a->/x. 

35.  12  Vf.       37.  2aV^. 

39.  5V2.  41.  2-^. 


32.  7V3.. 

33.  10  V5. 

38.  3V2. 

Hint.    3^2=  \/27  .  \^. 


43. 


40.  2V3. 


42.  3V4. 


44.  (a-\-x) 


The  multiplication  of  binomial  or  of  polynomial  radical 
expressions  of  the  same  order  involves  no  new  principle. 


RADICALS  263 


EXAMPLE 

Multiply        3V5--4V3by2V54-V3. 
Solution.    3  V5  -  4  V3 

2V5  +   Va 


30      -8Vl5 

+  3  VTE  - 12 
30      -  5  Vl5  - 12  =  18  -  5VT5. 


EXERCISES 


Perform    the    indicated    multiplication    and    simplify   the 
results : 

1.  ( V2  -  3)(3  V2  +  5).  3.  (4  Vt  -  3)'. 

2.  (2  V5_4)(3V5-f  3).  4.  (2V7-3V2y. 

5.  (3V2+V3)(2V3- V2). 

6.  (2V5-3V2)(3V5  +  2V2). 

7.  (  V5  -  V3  +  V2)(  V5  -  V3  +  V2). 

8.  (3  V2  +  2  V3  4- V30)(  V5  +  V3  -  V5). 

9.  (72V2-2)(i2V2-3). 

10.  {rV3-rV2){rV3-\-2E-V2). 

11.  (7e-fV2)(ie-fV2). 

(|,|V3)-.        -«--(f-fV2)' 


12. 


(V6-1)'.  16.  ^|«'-(f-fV3)' 


KB 


264  FIRST  COURSE  IN  ALGEBRA 

18.  The  hypotenuse  of  a  right  triangle  is  R  and  one  leg  is 

H  _  —Vs.    Find  the  other  leg. 

R 

19.  One  leg  of  a  right  triangle  is  -^(  V5  —  l)  and  the  hy- 
potenuse is  R,    Find  the  other  leg. 


R 


20.  The    legs    of    a    right    triangle   are   R  V2  +  V2   and 
\2  —  V2.    Find  the  hypotenuse  and  the  area. 

21.  Show  by  substituting  and  simplifying  that  2  +  2  VS  is 
a  root  of  the  quadratic  equation  x^  —  4  cc  =  8. 

22.  Does  \  —  \  V4I  satisfy  a;^  -7x  +  2  =  0  ? 

112.  Radical  equations.  An  irrational  or  radical  equa- 
tion in  one  unknown  is  an  equation  in  which  the  unknown 
occurs  in  a  radicand.  The  solution  of  the  simpler  types  of 
radical  equations  depends  mainly  on  a  knowledge  of  mul- 
tiplication of  radicals.  The  ability  to  square  or  to  cube 
each  member  of  an  equation  and  the  exercise  of  a  little 
judgment  in  transposing  at  the  proper  time  are  sufficient  to 
solve  any  of  the  equations  which  arise  in  elementary  work. 

The  necessity  of  checking  all  results  obtained  cannot  be 
too  strongly  emphasized.  The  pupil  should  remember  that 
an  answer  is  a  root  on  the  one  condition  that  it  satisfies  the 
original  equation, 

EXERCISES 

Solve  for  x  and  check : 


1.   V2x-7=3.  6.   Vaj-3-6  =  0. 

Hint.    Squaring  each  member,       Hint.   Transpose  before  squaring. 
2x- 7  =  9,  etc. 


2. 

VSa; 

-5  = 

=  5. 

3. 

^x- 

-6  = 

Vs. 

4. 

V2a; 

+  3 

=  V7. 

5. 

2V^ 

-1  = 

=  V6. 

7. 

2V3ic+l-8  =  0. 

8. 

2V5aj-4  +  3=ll. 

9. 

^^.-1=2. 

10. 

v'2a;-f  1=3. 

EAI 
-6  =  4, 

>ICALS 

+  4  = 

265 

11. 

-^40.- 

14.  2-y/Sx-9 

0. 

12. 

^2x- 

-1  +  5  =  0. 

15.  3Vaj-2  = 

■.■\^2x 

+  3. 

13. 

3-\/2^ 

3-V^ 

T.    Squai 
Qsposing, 
iding  by 
aring  aga 

+  7=6. 

16.  2ViK  +  3  = 

=  Va;4 

=  cc. 

=  -12. 
=  2. 
=  4,  etc 

-18. 

17.  3V2a^-6 

18.  V2:c-1- 
HiNT.    Transpose, 

=  2  V:k  +  4. 
.■^2-x  =  0, 
then  square. 

19.  V4:r-3- 

-^3x  +  5  =  0. 

-  V5  cc  +  7  =  0. 

-  3  V2  +  X  =  0. 

9_6Vx  +  3  +  xH-3  = 

-6Vx4-3r 

Vx  +  3  = 

X  +  S-- 

22. 

HiN 

Tra] 
Div: 

Squ 

20.  ^6x-5  - 

21.  2V3CC-3 
c  +  3  =  Vx. 
•ing  each  member, 

-6, 
in. 

23. 

2x  +  5  =  -V2x. 

24.  7- V3a;  +  7- V3a;  =  0. 
Hint.    7  —  VSx  =  V3x  +  7,  etc. 


25.  -VSx  -  5  +  ■\^3x  +  7=6. 

26.  V^  +  V2-Va;  +  2  =  0. 


27.  Vaj  +  1  -  V2aj  -  3  =  V3aj  -  2. 


Hint.    Squaring,  x  +  1  -  2 V(x  +  l)(2x-3)  +  2x  —  3  =  3x-2. 
Transposing,  2  V(x  +  1)  (2  x  —  3)  =  0,  etc. 


28.  Vx-3  +  V2x  +  4  =  V3a;  +  1. 

29.  Vx  +  3  +  Vx  -  4  =  V4x  —  3. 

30.  Va;  +  4  +  Vx  -  3  =  V4a;  -  3. 


Vaj-3       Va;-4  ^^     Vcc  -  3       Vx  -  5 

Vic  +  l       ■Vx-2  '   Vx  +  2       Vic-l 


266  FIEST  COUESE  IN  ALGEBEA 

113.  Division  of  radicals.  It  is  frequently  necessary  to 
find  the  approximate  value  of  an  expression  which  involves 
division  by  a  radical  expression.    Thus   2  -h  V3,   V3  -r-  VS, 

=r9  and  — ^= ~  are  types  which  often  occur. 

2_V3  V5+V2 

To  find  the  approximate  value  of  2  -^  VS  we  may 
extract  the  square  root  of  3  to  several  decimal  places 
and  then  divide  2  by  the  approximate  root  obtained. 
Both  of  these  processes  are  long  and  one  of  them  is 
unnecessary.   For,  writing  2  -^-  V3  as  a  fraction  and  multiply- 

ing  both  terms  by  V3  gives To  find  the  approximate 

o 

value  of  this  last  fraction  requires  but  one  long  operation. 
Similarly,  the  process  of  finding  the  approximate  value 
of  V7-J-(V7  — V2)  involves  three  rather  lengthy  operations 
—  the  extracting  of  two  square  roots,  and  one  long  divi- 
sion. The  labor  of  two  of  these  operations  can  be  avoided 
in  a  manner  similar  to  that  shown  above. 

114.  Rationalizing  factors.  One  radical  expression  is 
the  rationalizing  factor  for  another  if  the  product  of  the 
two  is  rational. 

A  rationalizing  factor  of  Vs  is  Vs,  for  Vs  •  Vs  =  3. 

For  -V2  a  rationalizing  factor  is  \^4,  since  -V2  •  V4  =  S/s  =  2. 

Similarly,  Vz  —  V2  is  a  rationalizing  factor  of  V?  +  V2,  as  their 
product  ( Vt  -  V2)  ( Vt  +  V2)  =  7-2  =  5. 

.  In  like  manner   (3_V5  -  2  Vs)  (3  Vs  +  2  V3)  =  45  - 12  =  33. 
Therefore  3  V5  —  2  V3  is  a  rationalizing  factor  of  3  V5  +  2  V3. 

The  binomial  radicals  of  the  last  two  illustrations  are  of 
the  general  types  Va  +  Vb  and  Va  —  V5.  Such  binomials 
are  called  conjugate  radicals  and  either  is  a  rationalizing 
factor  of  the  other.  If  a  and  h  are  rational,  the  product 
(Va  -f  V6)  (Va  —  \/6),  or  a  —  6,  is  also  rational. 


1. 

V5. 

6. 

V27. 

2. 

3  Ve. 

7. 

^2. 

3. 

2V7.. 

8. 

-^. 

4. 

v§. 

9. 

2-^5. 

5. 

V32. 

10. 

^/16. 

16. 

4V3-V2. 

17. 

2V5H-TV6. 

18. 

■\/x  —  Va. 

19. 

V3a4- Vx. 

20. 

V»-2ic. 

RADICALS  267 

ORAL  EXERCISES 

Determine  a  rationalizing  factor  for  each  of  the  following 
expressions  and  find  the  product  of  the  given  expression  and 
this  factor : 

11.  -v^25. 

12.  V2  +  3. 

13.  V3-V2. 

14.  3  +  V7. 

15.  3V2-5. 

Direct  division  of  similar  radicals,  coefficient,  by  coeffi- 
cient and  radicand  by  radicand,  is  often  possible. 

Thus  V6-^V2=V3. 

And  12  VlO  ^  2  V5  ==  6  V2. 

But        V7iv3  =  :^  =  :^l^=^. 

V3     V3  •  V3        3 

If  direct  division  cannot  be  exactly  performed,  we  use  the 

Rule.    Write  the  dividend  over  the  divisor  in  the  form  of  a 

fraction.    Then  multiply  the  numerator  and  denominator  &f 

the  fraction  hy  the  rationalizing  factor  of  the  denominator 

and  simplify  the  resulting  fraction, 

EXERCISES 

Perform  the  indicated  division : 

1.  Vl0-r-V2.  4.   V7-f-V3. 

2.  Vl8^V3.  5,  2V6-3V2. 

3.  V5-^V2. 
Solution.    V5-^V2. 

V5       V5 .  'V2      VlO 


6.  Vl0-r-V3. 

7.  2V7^V2. 


V2    V2.V2      2  8.  Ve-^-Vis. 


268  FIEST  COURSE  IN  ALGEBEA 

9.  6-2V2.  12.   V6-VI8-V2. 

10.  8VI5-4V5.  13.  (Vi2-V24)-3V3. 

11.  3V2-f-15V8.  14.  (Ve- V9+18)-i-2V2. 

15.  8--(3  +  V7). 

_8  8(3 -V7) 


Solution.    8  -^  (3  +  V7) 

CiA  O  ^  /n  _ 

■=12-4V7, 


3+V7      (3+V7)(3-V7) 
24-8V7 


9  -7 

16.  9--(3-V2).  2V3 

17.  V5^(V5-V2).  '   V3+V2' 
Hint.  ^^     4V5+V3 

V5  V5(V6  +  V2)  *     V5-V3' 

— = =  =  — = -^= = —     _  ,  etc.  f—         i- 

V5-V2     (V5-V2)(V6  +  V2)    .  00    •V3+V7 


V7-2 


Find  to  three  decimals  the  value  of : 


21.  5  +  3V2.  ^^     V2^  26 

22  "     ' 


7-3V5.  '   V3'  *   V3-V2 

7+V6  25     V5+I.  27.  3V8-V7. 

^^*        3      *  *       V5  28.  V3-V2. 

Simplify : 

29.   Va-j-V^.       30.    Vs^^^Vox^.       31.  2V^^3V^. 

Biographical  Note.  Frangois  Vieta.  The  reason  that  algebra  is  a 
universal  language  which  does  not  depend  entirely  on  the  nationality  of 
the  writer  lies  in  the  fact  that  the  symbols  used  to  indicate  the  various 
operations  and  relations  are  widely  understood  and  adopted.  This  has 
not  always  been  the  case,  and  for  a  long  time  during  the  early  history 
of  the  subject  there  was  no  accepted  notation  in  algebra,  but  each  man 
used  any  symbol  that  suited  him.  One  of  the  men  who  did  most  to  estab- 
lish a  fixed  notation  was  Francois  Vieta  (1540-1603),  a  French  law^^er 
who  studied  and  wrote  on  mathematics  as  a  pastime.  He  was  in  public 
life  during  his  whole  career  and  was  well  known  for  his  ability  to 
decipher  the  hidden  meaning  of  dispatches  captured  from  the  enemy- 


FRANCOIS  VtETA 


EADICALS  269 

It  was  he  who  established  the  use  of  the  signs  +  and  —  for  addition 
and  subtraction,  which,  to  be  sure,  had  been  used  before  his  time,  but 
were  not  generally  accepted.  He  also  denoted  the  known  numbers  in 
an  equation  by  the  consonants,  B,  (7,  D,  etc.,  and  the  unknowns  by 
the  vowels.  A,  E,  I,  etc.  He  recognized  the  existence  of  negative  roots 
of  equations,  but  rejected  them  as  absurd. 

To  denote  the  second  and  third  powers  of  the  unknown,  he  used  the 
letters  Q  (quadratus)  and  G  (cubus)  respectively.  Instead  of  using  the 
sign  =  he  wrote  aeq.  (aequalis  or  aequatur).  Thus  Vieta  would  have 
written  the  equation  x^  —  8  x^  +  16  x  =  40  in  the  form 

lC-8Q4-16iVaeq.40. 

Before  the  time  of  Vieta  this  equation  would  have  been  written  in  a 
much  more  primitive  notation.  For  instance,  with  writers  only  a  little 
earlier  it  would  appear  as 

Cubus  m  SCensusp  16  rebus  aequatur  40. 

It  is  easily  seen  that  operations  on  equations  in  this  form  would  be  very 
hard  to  perform. 

Vieta  is  further  distinguished  as  being  the  first  man  to  obtain  an  exact 
numerical  expression  for  the  number  tt,  which  occurs  in  geometry.  His 
form  of  expression  calls  for  an  infinite  number  of  operations  which,  of 
course,  could  never  be  performed,  but  the  further  one  proceeded,  the 
closer  would  be  the  approximation  obtained.  In  a  certain  sense  the 
familiar  sign  V  implies  an  infinite  number  of  operations,  for  one  can 
never  go  through  the  process  of  extracting  the  square  root  of  2,  for 
instance,  and  come  out  even.  Vieta's  method  of  denoting  tt  was,  how- 
ever, more  involved  than  this  and  made  use  of  complicated  irrational 
fractions. 


CHAPTER  XXI 

QUADRATIC  EQUATIONS 

115.  Solution  by  completing  the  square.  The  quadratic 
equation  was  defined  on  page  137.  Before  taking  up  the 
work  that  follows,  the  student  should  review  the  method 
of  forming  trinomial  squares  given  on  page  121. 

ORAL  EXERCISES 

What  terms  should  be  added  in  order  to  make  the  following 
expressions  perfect  squares  ? 

1.  x^+2x-\-?  5.  0^2 4- 18 :r  4-  ?  9.  x''-5x-}-? 

2.  a^'+lx-f?  e.  x'^-lSx-}-?  10.  x'^-{-7x  +  ? 
Z.x^-10x-\-?  7.x^-\-3x  +  ?  11.  2^2_|^_p9 
^.x^-6x  +  ?  S.x^'  +  x-^?  12.  t''-{-^t-{-? 

EXAMPLES 

1.  Solve  i:b2  + 4  0^-21  =  0.  (1) 

Solution.    Transposing,    :r2  +  4  a:  =  21.  (2) 

Adding  4  to  each  member  of  (2), 

a;2+4:r +  4  =  25.  '       (3) 

Then  (x  +  2)^  =  52.  (4) 

Extracting  the  square  root  of  each  member  of  (4), 

a:  +  2  =  ±  5.  (5) 

Whence  a:  =  -2  +  5  =  3, 

and  X  =  —  2  —  5  =  —7^ 

27Q 


QUADRATIC  EQUATIONS  271 

Check.    Substituting  3  for  x  in  (1),  9  +  12  -  21  =  0,  or  0  =  0. 
Substituting  -  7  for  x  in  (1),  49  -  28  -  21  =  0,  or  0  =  0. 

2.  Solve  3^2- lOx- 8  =  0.  (1) 

Solution.    Transposing,                       Zx'^  —  10x  =  8.  (2) 

Dividing  (2)  by  the  coefficient  of  x,  x^ — -  =  - .  (3) 

o  o 

Adding  (—  g)'^  to  each  member  of  (3), 

3\3/       39        9  ^  ' 

Then  {^-if=af-  (5) 

Extracting  the  square  root  of  each  member  of  (5), 

^'  -  8  =  ±  5- 

Whence  x  =  §  i  3  =  4  and  —  §. 

Check.    Substituting  4  for  x  in  (1), 

3-42-10.4-8  =  0. 

48  -  40  -  8  =  0,  or  0  =  0. 
Substituting  —  §  for  x  in  (1), 

3(-§)2-10(-S)-8  =  0. 
I  +  2^0  _  8  =  0. 

2^4:  _  8  =  0,  or  0  =  0. 

The  method  of  solving  a  quadratic  equation  in  x  illus- 
trated in  the  preceding  examples  may  be  stated  in  the 

Rule.  Transpose  so  that  the  terms  containing  x  are  in  the 
first  member  and  those  which  do  not  contain  x  are  in  the  second. 

Divide  both  members  of  the  equation  by  the  coefficient  of  x^ 
unless  that  coefficient  is  -\- 1, 

Then  add  to  both  members  the  square  of  one  half  the 
coefficient  of  x  (in  the  equation  just  obtained^,  thus  making 
the  first  member  a  perfect  trinomial  square. 

Rewrite  the  equation,  expressing  the  first  member  as  the  square 
of  a  binomial  and  the  second  member  in  its  simplest  form. 


272  FIRST  COURSE  IN  ALGEBRA 

JEJxtract  the  square  root  of  both  members  of  the  equation 
a7id  write  the  sign  ±  before  the  square  root  of  the  second 
member^  thus  obtaining  two  linear  equations. 

Solve  the  equation  in  which  the  second  member  is  taken 
with  the  sign  +  and  then  solve  the  equation  in  ivhich  the 
second  member  is  taken  with  the  sign  — .  The  results  are  the 
roots  of  the  quadratic. 

Check,  Substitute  each  result  separately  in  place  of  x  in 
the  original  equation.  If  the  resulting  equations  are  not 
obvious  identities^   simplify  each  until  it  becomes  one. 

If,  after  the  equation  has  been  simplified  in  accordance 
with  the  first  two  directions  of  the  foregoing  rule,  the 
term  in  the  first  power  of  x  is  lacking,  that  is,  the  equa- 
tion is  in  the  form  7?=b/it  is  often  called  a  pure  quadratic. 
Such  an  equation  may  be  solved  immediately  by  the 
extraction  of  the  square  root  of  each  member. 

Thus,  if  5  a;2  -  3  =  12,  then  b  x' =  15,  x^  =  3,  and  x  =  ±  V3. 

EXERCISES 

Solve  by  completing  the  square,  where  necessary,  and  check 
as  directed  by  the  teacher. 

11.  2cc''-9;r +  4  =  0. 

12.  2?^2_3^^9 

13.  3^2-5-70. 

14.  c«2  4-8  =  4x2-40. 

15.  bx  =  6;:c2-14. 

16.  .-If -1=0. 

17.  iM''-13t-{-^  =  0. 

18.  42  4-2cf-  =  -19a^. 


1. 

x'- 4.x -32  =  0. 

2. 

x^-2x-lb  =  0. 

3. 

x'=l^. 

4. 

x^-25  =  0. 

5. 

^2_7^_18  =  0. 

6. 

x(x-\-A)-S(x  +  4:)  =  0. 

7. 

y  +  6  =  f. 

8. 

4.x'-4:X-3  =  0. 

9. 

^2_5i^^^2i  =  0. 

10. 

l^-x'  =  2x. 

QUADRATIC  EQUATIONS  273 

19.  Why  is  not  equation  (5),  Example  1,  page  270,  written 
with  the  sign  ±  before  each  member  ? 

20.  2cc2-5x-f  1=0.  (1) 
Solution.  Transposing,  2 x'-^  —  6x  =  —1.  (2) 
Dividing  each  member  of  (2)  by  2, 

a:2-f2r  =  -4.  (8) 

Adding  (—  j)^  to  each  member  of  (3), 

:c^-ix  +  (-i)^  =  -,i  +  -ft  =  H-  (4) 

Then  (X  -  if  =  H-  (5) 

Extracting  the  square  root  of  each  member  of  (5), 

x-i=^±V\i=±i^'T7.  (6) 

Whence                          '  x=^±  i Vl7.  (7) 

Now                          ^VT7=  J  (4.123+)  =  1.031-.  (8) 

Then                  |  +  iVl7  =  1.25  +  (1.031 -)-=  2.281 -.  (9) 

Also                   I  -  i  Vl7  =  1.25  -  (1.031  -)  =  .219  + .  (10) 

Check.  Since  (9)  and  (10)  are  hot  the  exact  values  of  x,  they  will, 
if  substituted  for  x  in  (1),  make  its  first  member  nearly  but  not 
quite  zero.  An  exact  check  on  the  radical  forms  of  the  roots  can  be 
obtained  by  substituting  from  equation  (7)  in  equation  (1). 

The  check  may  be  shortened  by  substituting  both  roots  at  the 
same  time,  as  follows  : 

Substituting  f  ±  J  Vl7  for  a:  in  2  a:^  —  5  x  +  1  =  0, 
2  (I  ±  i  Vriy  -  5(f  ±  i  V17)  +  1-0. 

2(ft±|Vn+  H)-5(4±iVi7)  +  i-o. 
V'  db  I  A'/rr  +  V-  -  ¥  +  I  ^^  + 1  -  o. 

The  radical  terms  vanish  because  the  two  upper  signs  before 
them  must  first  be  taken  together,  and  then  the  two  lower  signs. 

Therefore         ^/  +  ^f  -  %^  +  |  =  0,  or  ^^Q  -  ^^^  =  0. 

In  quadratic  equations  like  the  preceding  the  radical  forms 
of  the  roots  are  often  sufficient ;  at  other  times  values  to  two 


274  FIEST  COURSE  IN  ALGEBRA 

or  three  decimal  places  are  necessary.    Unless  otherwise 
directed,  obtain  only  the  radical  forms  of  irrational  roots. 

In  the  following  ten  exercises  obtain  correct  to  three  deci- 
mal places  the  values  of  any  radical  answers  which  may  occur  : 

21.  p^-^10p-{-17=0.  27.  x''-6-V3x=  -9. 

22.  a;2  +  2 j:c  -  4  =  0.  2S.  q^  -  3q  V2 -\- A  =  0. 

23.  cc2-2.a^-3i  =0.  29.  x^ -\- 5  =  6  -  4:X -Sx'', 

24.  3x^-6x-i-l=0,  1 

25.  9x-  =  5cc^-2.  ^^'  ^-^J^r^^' 
26.1-4^2^2?;.  31.  cc^-4x2  +  3  =  0. 

This  is  not  a  quadratic  equation,  but  many  equations  of 
this  form  can  be  solved  by  completing  the  square. 
Solution.  a;4  -  4  a;2  +  3  =  0. 

a;4-4a:2  +  4  ==-3  +4=1. 
a;2  -  2  =  ±  1. 

a;2  =  3  and  1. 

a:=iV3  and  ±1. 
Check  as  usual. 

32.  i^*- 5x^  +  4  =  0.  35.  a:* -10^^2  +  24  =  0. 

33.  9x^-13a^2^4  =  0.  36.  4.x^ -l^x" -\-3  =  0. 

34.  9x^-82x2  +  9  =  0.  37.  16x^-81x2  +  5  =  0. 

Note.  The  student  has  undoubtedly  noticed  that  quadratic  equa- 
tions have  two  roots.  Mathematicians  attempted  for  many  years  to 
prove  that  cubic  equations  always  have  three  roots  and  that  equa- 
tions of  the  fourth  degree  have  four  roots.  This  was  finally  done  in 
the  seventeenth  century  by  the  Italians  Cardan  and  Ferrari. 

That  any  equation  in  one  unknown  has  a  number  of  roots  equal 
to  its  degree  was  first  proved  by  K.  F.  Gauss.  He  gave  three  distinct 
proofs  of  this  fact,  although  no  one  before  his  time  had  been  able  to 
prove  it  at  all.  Since  the  time  of  Gauss  hundreds  of  proofs  have 
been  given  by  mathematicians  in  all  parts  of  the  world. 

The  mathematical  researches  which  have  been  stimulated  and 
carried  to  a  high  degree  of  completeness  by  the  study  of  this  problem 


QUADRATIC  EQUATIONS  275 

have  been  very  numerous.  This  situation  illustrates  the  important 
fact  that  when  a  scholar  takes  up  a  scientific  study  with  the  intention 
of  reaching  a  definite  result,  the  indirect  results  of  his  investigations 
are  frequently  even  more  important  than  the  solution  of  the  main 
problem  which  he  had  set  himself. 

Few  scientists  possess  the  vision  to  determine  surely  at  the  begin- 
ning of  an  investigation  all  of  the  directions  in  which  it  may  lead. 
The  distinction,  however,  between  a  great  man  and  a  little  one  largely 
lies  in  the  instinct  which  the  great  man  has  to  treat  subjects  which 
prove  to  be  fruitful. 

REVIEW  EXERCISES 

Solve,  and  check  as  directed  by  the  teacher : 

1.  12x^-^7x  =  5. 

2.  (5x-2)(x~6) 

=  (2^-12)(2a^-4). 

3.  4.x''  +  15x  =  -9. 

4.  ,T  + 1  =  20  0^1 

5.  25^'^-20^-12:==0. 

6.  6x-\-x^-j-3 

=  2  -  3  x'  +  10  a^. 

7.  (3x-{-6)(x-3) 
2). 


x+2      x+1 

21.  (3x-2)(.T-3) 

•  =(x-{- 1)  (x  +  4). 

22.  x^-10x''-i-16  =  0. 

23.  x^-Sx^-\-15  =  0. 

24.  6:^^-19x2  +  15  =  0. 

25.  8aj^-18a;2+9  =  0. 


=  (2x  +  l)(x 

8. 

i+*-f-»- 

9. 

hi-h- 

10. 

3a!       1        1 
2    '^2      3x 

11. 

3 

12. 

X  —  b       2 

13. 

2          3._0 

t-2       2 

14. 

X          5x3 
5-x   '    12  ~  2 

15. 

X          5      2  —  x 

x-2      2          X 

16. 

2             1             15 

s-2      5+2"       8 

17. 

3+x       1        h-x 
4  -f-  X      12       6  -  X 

18. 

l-\-x      1      3-x 
2+x      6      5-x 

19. 

x+2      x-1 

x-4   '   x-t-5 

20. 

x-2      x-l_^^ 

276  FIRST  COURSE  IN  ALGEBRA 

PROBLEMS 

(Reject  all  answers  which  do  not  satisfy  the  conditions  of  the  problems.) 

1.  The  square  of  a  certain  number  plus  twice  the  number 
itself  is  8.    Find  the  number. 

2.  If  from  twice  the  square  of  a  certain  number  the  num- 
ber itself  be  taken,  the  remainder  is  28.    Find  the  number. 

3.  Find  two  consecutive  odd  numbers  whose  product  is  675. 

4.  Find  three  consecutive  even  numbers  whose  sum  is  |  the 
product  of  the  first  two. 

5.  A  rectangular  field  is  14  rods  longer  than  it  is  wide.  Its 
area  is  20  acres  (1  acre  =  160  square  rods).  Find  the  dimen- 
sions of  the  field. 

6.  The  sum  of  a  certain  number  and  its  reciprocal  is  2|^. 
Find  the  number. 

7.  The  area  of  a  triangular  field  is  4|-  acres.  The  base  is 
36  rods  longer  than  the  altitude.  Find  the  base  and  the  altitude. 

8.  Two  square  fields  together  contain  40  acres.  A  side  of 
one  is  16  rods  longer  than  a  side  of  the  other.  Find  the  side 
of  each. 

9.  The  hypotenuse  of  a  right  triangle  is  41  feet.  One  leg 
is  31  feet  shorter  than  the  other.    Find  the  legs. 

10.  One  leg  of  a  right  triangle  is  -|  as  long  as  the  other. 
The  hypotenuse  is  30.    Find  the  legs. 

11.  The  area  of  a  square  in  square  feet  and  its  perimeter  in 
inches  are  expressed  by  the  same  number.    Find  the  side. 

12.  The  dimensions  of  a  certain  rectangle  and  the  longest 
straight  line  which  can  be  drawn  on  its  surface  are  represented 
in  inches  by  three  consecutive  numbers.    Find  its  dimensions. 

13.  The  edges  of  two  cubical  bins  differ  by  one  yard.  Their 
volumes  differ  by  127  cubic  yards.    Find  the  edge  of  each  bin. 


QUADRATIC  EQUATIONS  277 

14.  The  rates  of  two  trains  differ  by  6  miles  per  hour.  The 
faster  requires  1  hour  less  time  to  run  252  miles.  Find  the  rate 
of  each  train. 

15.  An  automobile  made  a  round  trip  of  180  miles  in  11  hours. 
On  the  return  the  rate  was  increased  3  miles  per  hour.  Find  the 
rate  each  way. 

16.  A  page  of  a  certain  book  is  2  inches  longer  than  it  is  wide. 
The  printed  portion  covers  half  the  area  of  the  page  and  the 
margin  is  1  inch  wide.   Find  the  length  and  width  of  the  page. 

17.  The  price  of  oranges  being  raised  10  cents  per  dozen, 
one  gets  6  fewer  oranges  fo^^  a  dollar.    Find  the  original  price. 

18.  Two  pumps  together  can  fill  a  standpipe  in  30  minutes. 
One  pump  alone  requires  32  minutes  less  time  than  the  other. 
Find  the  time  each  requires  alone. 

19.  How  long  will  it  take  a  stone  to  reach  the  ground  if 
dropped  from  an  elevation  of  1600  feet  ?  (See  Exercise  45, 
p.  17.) 

20.  A  man  drops  a  stone  over  a  cliff  and  hears  it  strike  the 
ground  below  61  seconds  later.  If  sound  travels  1152  feet  per 
second,  find  the  height  of  the  cliff. 

116.  Quadratic  equations  with  literal  coefficients.  Such 
equations  are  solved  as  in  the  following  example. 

EXAMPLE 
Solve  for  x  the  equation  x^  —  ^  ax  —  1  a^  =  0,  and  check  the 
result. 

Solution.    Transposing,  x^  —  Q  ax  =  7  a^. 

Completing  the  square,  x'^  —  6  ax  +  9  a^  —  16  a^. 

Then  {x  -  3  af  =  16  ciK 

Extracting  the  square  root,  a;  —  8  a  =  ±  4  a. 

Whence,  transposing  and  combining,  x  —  1  a,  and  —  a. 

Check.    (7  a)2-  6  a  .  7«  -  Ici^  =  49  a^  _  40  a^  -  7^2  :^  0,  or  0  =  0. 

^^  a)2_  (]  ^J^^  ^>)  -  7(72  r^  r/2  +  6  fl2  -  7  Q^  -    0,   OF  0  =  0. 


278  FIEST  COURSE  IN  ALGEBRA 

EXERCISES 

Solve  for  x  and  check : 

1.  x''-^2hx  =  Z IP'.  9.  cc"  -  12 a  =  3 ax  -  4a;. 

2.  x"-  -\-^cx=.h cl  10.  2x''-2>hx-2}v'=.  0. 

3.  ^2  -  8  coj  -  9  c^  =  0.  11.  lOcc^  -  3  a^  -  ax, 

4.  x'  +  «^^  -  2  a'  =  0.  12.  2  x^  -  5  Ao;  4-  2  A^  =  o. 

h.  x"  —  2ax^\  —  (T,  13.  ai:c^  —  5 cc  —  2 ax  + 10  =  0. 

6.  x'  _  a^  =  0.  14.  ex"  +  2  X  =  3  ex  +  6. 

7.  aV  =  J/.  '  15.  ax^  -f  a  =  a^x  +  x. 

8.  x^  +  Z^x  =-  2  b\  16.  cx^  +  ax  =  2  ex  +  2  a. 

History  of  the  quadratic  equation.  Though  the  development  of  the 
method  of  solving  quadratic  equations  is  closely  connected  with  the 
general  growth  of  algebra,  yet  it  is  possible  to  indicate  rather  briefly 
the  most  important  steps  in  the  process. 

The  first  writer  on  formal  algebra  was  Diophantos,  who  lived 
at  Alexandria,  in  Egypt,  about  a.d.  275.  Most  of  his  work  that  is 
preserved  is  devoted  to  the  solution  of  problems  that  lead  to  equa- 
tions. So  far  as  we  know  he  was  the  first  to  indicate  the  unknown 
number  by  a  single  letter,  in  this  respect  being  far  in  advance  of 
many  mathematicians  who  lived  much  later.  It  is  a  little  remarkable, 
in  fact,  that  so  able  and  original  a  man  as  Diophantos  should  have 
exerted  so  little  influence  on  his  successors.  He  solved  his  quadratic 
equations  by  a  method  not  unlike  that  of  completing  the  square,  but 
his  imperfect  knowledge  of  the  nature  of  numbers  made  it  impossible 
for  him  to  understand  the  entire  significance  of  the  process.  Though 
he  made  every  effort  not  to  consider  equations  whose  roots  were  not 
positive  integers,  sometimes  they  would  creep  in,  and  under  such  cir- 
cumstances, when  his  method  led  him  to  a  negative  or  irrational  root, 
he  rejected  the  whole  equation  as  absurd  or  impossible.  Even  when 
both  of  the  roots  were  positive  he  took  only  the  one  afforded  by  the 
positive  sign  in  the  formula  for  solving  a  quadratic. 

The  difficulties  of  Diophantos  are  typical  of  those  encountered  by 
mathematicians  for  the  next  fifteen  hundred  years.  The  difficulty 
lay  not  in  finding  a  formal  method  of  solving  the  equation,  but  in 
understanding  the  result  after  it  was  obtained.    The  meanings  of 


QUADRATIC  EQUATIONS  279 

negative  and  of  imaginary  numbers  have  been  two  of  the  most 
difficult  of  all  mathematical  ideas  for  men  to  grasp. 

Five  or  six  hundred  years  later  the  Hindus  devised  a  general  solu- 
tion of  the  quadratic,  but  their  chief  advance  over  Diophantos  lay 
in  the  fact  that  they  did  not  regard  an  equation  whose  roots  were 
negative  as  necessarily  absurd,  but  merely  rejected  the  negative  result 
with  the  remark,  "It  is  inadequate ;  people  do  not  approve  of 
negative  roots."  The  Hindus,  however,  did  realize  that  a  quadratic 
equation  sometimes  has  two  roots,  a  fact  that  Diophantos  never 
comprehended. 

No  material  gain  in  the  understanding  of  the  solutions  of  the 
quadratic  can  be  found  until  the  seventeenth  century.  The  keenest 
mathematicians  of  the  sixteenth  century,  like  Cardan  and  Vieta, 
rejected  negative  roots,  though  by  this  time  irrational  roots  were 
admitted.  In  fact,  in  1544  Stifel,  a  German,  published  an  algebra  in 
which  irrational  numbers  are  included  among  the  numbers  proper. 
But  he  affirms  that  except  in  the  case  where  a  quadratic  equation 
has  two  positive  roots  no  equation  has  more  than  one  root.  It  was 
not  until  the  work  of  Descartes  and  Gauss  became  widely  known 
that  the  nature  of  the  roots  of  all  kinds  of  quadratic  equations  was 
completely  understood. 

117.  Systems  involving  a  linear  and  a  quadratic  equation. 
A  quadratic  equation  in  two  unknowns  contains  one  or  more 
terms  of  the  second  degree,  but  no  term  of  higher  degree 
in  those  unknowns.  Every  system  of  equations  in  two 
unknowns  in  which  one  equation  is  linear  and  the  other 
quadratic  can  he  solved  hy  the  method  of  substitution. 

EXAMPLE 

.  Solve  the  system  f^'  +  Z/'^^,  (1) 

\x-y=l.  (2) 

Solution.    Solving  (2)  for  a:,  a:  =  1  +  y.  (3) 

Substituting  1  +  ?/  f or  x  in  (1),  (1  +  yY  +  /  =  5.  (4) 

From  (4),  y2^y_2  =  0.  (5) 

Solving  (5),  2^  =  1  or  —  2. 


280  FIEST  COURSE  IN  ALGEBRA 

Substituting  1  for  y  in  (3),  x  =  \-\-l  =  2. 

Substituting  -  2  for  ?/  in  (3),  x  =  l-2  =  -l. 

fx  =  2  fx  =  -l 

Therefore  <  ^  and  <  ,^  are  the  two  sets  of  roots. 

f  (1)       4  +  1  =  5 
Check.    Substituting  2  for  x  and  1  for  ?/  in  k  ^^:J'  i  _  i' 

f  (1),       1  +  4  =  5, 
Substituting  —  1  for  x  and  —  2  for  ?/  in       <  ^^^^  o  _  i' 

The  similarity  between  this  method  and  that  for  the  sohition  of 
linear  systems  by  substitution  should  be  carefully  noted. 

The  method  of  the  preceding  solution  is  stated  in  the 

Rule,  Solve  the  linear  equation  for  one  unknown  in  terms 
of  the  other. 

Substitute  this  value  in  the  quadratic  equatioji  and  solve 
the  resulting  equation. 

Substitute  each  of  the  roots  of  the  quadratic  equation 
thus  found  in  the  linear  equation,  and  solve,  thus  obtaining 
two  sets  of  roots  of  the  shnultajieous  system. 

Check,  as  usual,  in  both  equations. 

EXERCISES 

Solve  the  following  systems,  pair  results,  and  check  each 
set  of  roots : 

1    x'  +  y  =  ^  x'  +  if  =  26,  4^  +  5./  =  6, 

'  2x-{-y  =  ^.  '  x  —  Si/  =  5.  *  joq  +  2  =  0. 

^x  +  ^j  =  5,  ^xy=12,  ^    4x-3y  =  80, 


a^2  +  2/'^13.        ^'  2x  +  y-10  =  {).      "'  xy+12  =  0. 

ic2  +  y  =  25,  x'  +  ^xy  =  2b,  3^+2^=5, 

3a:_42/  =  0.         '  2x-{-y=10.  AB+^  =  ^A, 

^Q    x'  +  ^y  =  5,  ^^    :r^j^?>xy  +  f  =  22, 

'  X  —  y  =  %.  '  2x  =  y. 

..     x'  +  f  +  y  =  4.Q,  x'  +  f-4.x-2y  =  2(i 

'  2x  +  y=10.  3x-y=10. 


QUADRATIC  EQUATIONS  281 

PROBLEMS 

(Reject  all  results  which  do  not  satisfy  the  conditions  of  the  problem.) 

1.  Find  two  numbers  whose  difference  is  3  and  the  differ- 
ence of  whose  squares  is  45. 

2.  The  sum  of  two  numbers  is  18  and  the  sum  of  their 
squares  is  170.    Find  the  numbers. 

3.  A  rectangular  field  is  39  rods  longer  than  it  is  wide  and 
its  area  is  10  acres.    Find  the  length  and  the  width. 

4.  The  difference  of  the  areas  of  two  squares  is  208  square 
feet'  and  the  difference  of  their  perimeters  is  32  feet.  Find  a 
side  of  each  square. 

5.  The  perimeter  of  a  rectangle  is  92  feet  and  its  area  is 
504  square  feet.    Find  the  length  and  the  width. 

6.  The  base  of  a  triangle  is  5  inches  longer  than  its 
altitude.  Its  area  is  1^^  square  feet.  Find  the  base  and  the 
altitude  of  the  triangle. 

7.  The  perimeter  of  a  rectangle  is  3  c  and  its  area  is  —  • 
Find  its  dimensions. 

8.  Do  positive  integers  differing  by  3  exist  such  that  the 
sum  of  their  squares  is  115  ?    If  so,  find  them. 

9.  If  a  two-digit  number  be  multiplied  by  the  sum  of  its 
digits,  the  product  is  576.  If  three  times  the  sum  of  its 
digits  be  added  to  the  number,  the  result  is  expressed  by 
the  digits  in  reverse  order.     Find  the  number. 

10.  The  annual  income  from  a  certain  investment  is  |48.  If 
the  principal  were  |200  more  and  the  rate  of  interest  1  %  less,  the 
annual  income  would  be  |2  more.  Find  the  principal  and  the  rate. 

11.  A  wheelman  leaves  A  and  travels  north.  At  the  same 
time  a  second  wheelman,  who  travels  50%  faster  than  the  first, 
leaves  a  point  3  miles  east  of  A  and  travels  east.  An  hour 
after  starting,  the  distance  between  them  is  17  miles.  Find 
the  rate  of  each. 


CHAPTER  XXII 

RATIO  AND  PROPORTION 

118.  Ratio.  The  ratio  of  one  number,  a,  to  a  second 
number,  h^  is  the  quotient  obtained  by  dividing  the  first 

by  the  second,  that  is,  -•    This  ratio  is  also  written  a:  h. 

It  follows  from  the  above  that  all  ratios  of  numbers  are 
fractions  and  that  all  fractions  may  be  regarded  as  ratios. 

Thus  -  r  -—  1 1  and  — —  are  ratios  as  well  as  fractions. 

2    2x    a  —  h  -y/5 

We  may  speak  of  the  ratio  of  two  concrete  numbers  if 
they  have  a  common  unit  of  measure.  The  ratio  of  5  feet 
to  3  feet  is  |,  the  common  unit  of  measure  being  1  foot. 
Obviously  no  ratio  exists  between  5  years  and  3  feet. 

If  we  say  a  piece  of  paper  contains  54  square  inches,  we 
are  expressing  by  the  number  54  the  ratio  of  the  surface  of 
the  paper  to  the  surface  of  a  square  whose  side  is  one  inch. 

Every  measurement,  then,  is  the  determination  of  a  ratio, 
either  exact  or  approximate. 

EXERCISES 

Simplify  the  following  ratios  by  writing  them  as  fractions 
and  reducing  the  fractions  to  their  lowest  terms : 

1.  6:9.      3.  5:10.     5.  20  a?:  12x1    7.  3  hours  :  150  minutes. 

2.  12:8.    4.  14:7.     6.  17i:3f.        8.   900  pounds  :  1^  tons. 

.    9.  Separate  45  into  two  parts  which  are  in  the  ratio  of  7:8. 

Hint.    Let  7x  =  one  part ;  then  8x  =  the  other. 

282 


EATIO  AND  PEOPOETION  283 

10.  Separate  121  into  two  parts  which  are  in  the  ratio  3 : 8. 

11.  Separate  152  into  two  parts  which  are  in  the  ratio  2^  :7. 

12.  What  number  added  to  both  terms  of  the  ratio  5  :  9 
gives  a  result  equal  to  the  ratio  36  :  54  ? 

13.  What  number  subtracted  from  both  terms  of  the  ratio 
11 :  12  gives  a  result  equal  to  the  ratio  35  :  40  ? 

14.  If  ir  is  a  positive  number,  which  is  the  greater  ratio: 

3         S+x^       X  2x     ^ 

or  * 


4  4:-\-x'    x+l         2ic+l  ' 

Hint.  Reduce  the  two  fractions  in  each  part  to  respectively  equivalent 
fractions  having  a  common  denominator,  and  then  compare  numerators. 

15.  From  your  answers  to  Exercise  14  state  the  change 
that  is  produced  in  the  value  of  a  proper  fraction  (sometimes 
called  a  ratio  of  less  inequality)  by  adding  a  positive  number 
to  both  of  its  terms. 

16.  If  X  is  a  positive  number,  which  is  the  greater  ratio: 

—  or  ?    or  c 

5  5  -{-  X  X  2x 

17.  From  your  answers  to  Exercise  16  state  the  change  that 
is  produced  in  thn  value  of  an  improper  fraction  (sometimes 
called  a  ratio  of  greater  inequality)  by  adding  a  positive 
number  to  both  of  its  terms. 

119.  Proportion.  A  proportion  is  a  statement  of  equality 
between  two  ratios. 

Thus  I  =  g,  I  =  yS^j,  and  ^J  =  f  are  proportions,  for  the  equality 
of  the  ratios  is  evident  in  each  case. 

The  four  numbers  1,  2,  3,  and  6  are  said  to  be  in  pro- 
portion, for  the  ratio  of  the  first  pair  equals  the  ratio  of 
the  second  pair.  In  general,  the  numbers  a,  J,  ^,  and  d 
are  in  proportion  ii  a:h  =  c:d.  (1) 

In  (1),  a  and  d  (the  first  and  fourth  terms)  are  called 
extremes,  and  b  and  c  are  called  means. 


284  FIEST  COUKSE  IK  ALGEBEA 

Since  a  proportion  is  an  equality  between  two  ratios 

(fractions),  it  is  therefore  an  equation.    Hence  any  operation 

which  may  he  performed  on  an  equation  may  he  performed  on 

a  proportion,    (See  Axioms,  pp.  39-40.) 

a      c 
Thus,  in   the  proportion  t  =  -^  both  members  may  be 

multiplied  by  hd^  giving  ad  =  he.  Here  the  first  member 
is  the  product  of  the  extremes  of  the  proportion,  and  the 
second  member  is  the  product  of  the  means. 

Therefore,  in  any  proportion  the  product  of  the  extremes 
equals  the  product  of  the  means. 

EXERCISES 
1,  Form  proportions  from  the  following  by  supplying  the 

.    .      ^  .  .  4      8     ...   15      3^.7       ?    ,,,  45      ? 

missing  terms  :  W^  =  ^;  (h)  20""?'  ^''^  21  ""  3  '  ^^60^1' 

12      2  50      5  ?       2  12      ? 

(e)  -  =  3;  (/)y  =  j;  (9)  13  =  9;  W  32  =  8' 


Solve  the  following  for  x : 


1  _    16      144 


2.  2|:9  =  a;^:12.  5.1:4=1:^-  8.  —  - — 


x-1        5  13        x+11  9.x:s  =  t:f. 

^'2x-S~9'  *•  3a;~4x-2'        10.  f:  x  =  t :/. 

-y,  A  /V»  LL*       J        \      S      =      T      '.     X, 

4.   (x  — 3):5  =  7:--        7.   -  =  77^*  in^  i 

^  ^  2  X       25  12,  f:x  =  x:l. 

120.  Fourth,  mean,  and  third  proportionals.  A  fourth 
proportional  to  three  numbers  is  the  fourth  term  of  a  pro- 
portion in  which  the  three  numbers  appear  in  the  order  in 
which  they  are  given  as  the  first,  second,  and  third  terms 
of  the  proportion  respectively. 

Thus  6  is  a  fourth  proportional  to  2,  3,  and  4,  for  §  =  |.  Again, 
20  is  a  fourth  proportional  to  3,  5,  and  12,  for  |  =  ^ . 


EATIO  AND  PEOPORTION  285 

In  general,  /  is  the  fourth  proportional  to  three  numbers 
a^  b,  and  c  if  a      c 

A  mean  proportional  between  two  numbers  is  the  second 
or  the  third  term  of  a  proportion  in  which  the  means  are 
identical. 

Thus  2  is  a  mean  proportional  between  1  and  4,  for  1:  2  =  2  :4. 
Also  —  2  is  a  mean  projDortional  between  1  and  4,  for  1 :  —  2  =  —  2  : 4. 

In  general,  m  is  the  mean  proportional  between  two  numbers 
a  and  h  if  am  .,  ,    ,— 

m      b 

A  third  proportional  to  two  numbers  is  the  last  term  of  a 
proportion  which  has  for  its  first  term  the  first  of.  the 
given  numbers  and  for  both  second  and  third  terms  the 
second  given  number. 

Thus  16  is  the  third  proportional  to  9  and  12,  for  /^  =  jl-  Again, 
25  is  the  third  proportional  to  4  and  10,  for  t^  =  it- 

In  general,  t  is  the  third  proportional  to  the  numbers 
a  and  b  if  a     b 

EXERCISES 
Find  a  fourth  proportional  to : 
1.  2,  5,  and  8.      2.  1,  5,  and  4.  5.  6,  8,  and  t 

Htxt     —  —  — 

6~/'  4.  6,  9,  and  12.  7.  -  a;,  x^,  and  2  iB« 

Find  the  mean  proportionals  between : 

8.  1  and  9.  .      9.  4  and  25.  i2.  |  and  2. 

1      ^  10.  9  and  \,  13.  "v^and  V50. 

m"^?*  11.   2V  ^^d  h  1^-    "^l  ^^^  ^^- 


286  FIRST  COURSE  IN  ALGEBRA 

Eind  the  third  proportionals  to : 

15.  1  and  2.  17.  3  and  -7.  21.  7  and  2  VTi. 

„         1     2  18.  2  and  V5.  22.  a;  and  -  2  cc. 

Hint.   -  =  -. 

^     ^  19.  3  and  V6.  23.  a  and  a^. 

16.  5  and  15.  20.  3  and  2  V6„         24.  c  and  V^. 

121.  Proportions  from  equal  products.  That  the  numbers 
which  occur  in  a  pair  of  equal  products  may  be  used  iv 
various  ways  as  the  terms  of  a  proportion  is  illustrated 
m  the  following: 

From  2  •  6  =  4  •  3  we  may  write  the  proportions  f  =  |,  |  =  5> 
1=1,  and  §  =  f .  Each  of  these  four  statements  is  a  proportion,  for 
in  each  case  the  equality  of  the  ratios  is  evident. 

In  general,  if  the  product  of  two  numbers  a  and  d  equals  the 
product  of  two  other  numbers  b  and  c,  one  pair  may  be  made 
the  extremes  and  the  other  pair  the  means  of  a  proportion ; 

that  is,  if  ad=bCy 

^.  a     c  a     b 

then  -  =  - ,  or  -  =  - . 

b     d  c     d 

Proof,   li  a'  d  =  b'  CIS  divided  by  bd,  we  obtain 

h/~  m'     ^^     b~  d'  ^  ^ 

li  a*  d  =  b  '  c  is  divided  by  cd,  we  obtain 

a  _  b 
c      d 

The  transformation  from  (1)  to  (2)  above  (where  the  means  are 
interchanged)  is  called  alternation. 

If  a-  d  =  b'  c  is  divided  by  ac,  we  obtain 

*  =  1  (3) 

a      c  ^ 

The  transformation  from  (1)  to  (3)  (where  the  fractions  are 
inverted)  is  called  inversion, 


(2) 


m 
n 

r 
s 

KATIO  AND  PEOPORTION  287 

EXERCISES 

Write  as  a  proportion  in  three  ways : 

1.2.3  =  6.1.  2.  3  .  8  =  4  .  aj.  3.  a  .  ^  =  c  .  1. 

Write  as  a  proportion  having  /  for  the  fourth  term  : 

4.  5  ./=  3.2.  5.  5  .  6  =  2  ./.  6.  /=  y  • 

Write  the  following  proportions  by  alternation : 

7-  1  =  12-         ^'-  =  T'  9.-  =  —         10. 

^        ^^  a       0  X       n 

Write  the  following  proportions  by  inversion : 

11.  1  =  ^.        12.  ?  =  |.        13.^  =  ^.        14.:^  =  ^. 

122.  Proportion  by  addition.  The  dividend,  or  numerator, 
in  a  ratio  is  called  the  antecedent,  and  the  divisor,  or  denomi- 
nator, is  called  the  consequent. 

If  in  the  proportion  |  =  |  we  add  each  antecedent  to  its 
consequent  and  divide  the  sum  thus  obtained  in  each  case 
by  the  consequent  of  its  ratio,  we  have 

2  +  34  +  6  5      10 

-3-==-6-'     ^^     3  =  T' 
which  is  a  proportion.    In  the  same  manner  we  obtain  from 
the  proportion  f  =  ^f  the  proportion  ^^-  =  ||^. 

In  general,  if  four  numbers  a,  ft,  c,  and  d  are  in  proportion^ 
they  are  in  proportion  by  addition;  that  is,  the  sum  of  the  first 
two  terms  is  to  the  second  term  as  the  sum  of  the  last  two  terms 
is  to  the  fourth  term. 

Proof.    Let  f  =  £.  (1) 

b      d 


Adding  1  to  each  member,  we  have 

7  +  1=-  + 1,     or     - 
0  a 

Here  (2)  is  said  to  be  obtained  from  (1)  by  addition, 


_4.1=-  +  1,     or     -^  =  -j-  (2) 


1. 

1  =  tV 

4       22 

2. 

i  =  /Y- 

*•  11  ~  a; 

3. 

il=ll- 

5.  x:y=1 

288  FIRST  COUfiSE  IN  ALGEBRA 

EXERCISES 

Write  by  addition  and  test  results  in  the  numerical  exercises : 

6.  5:7  =  c:d. 

7.f:s  =  t:l. 

2:3.  8.  A^:A^  =  r^:ri. 

a       e  a  +  b       c-{-d 

9.  If  -  =  -7  snow  that = 

0       a  a  c 

123.  Proportion  by  subtraction.  If  in  the  proportion 
^  =  2.1.  ^e  subtract  each  consequent  from  its  antecedent 
and  divide  the  remainder  thus  obtained  in  each  case  by 
the  consequent  of  its  ratio,  we  have 

7—5      21  —  15        2       6 

— - —  =  — -— — ->  or  -  =  —  9  which  is  obviously  a  proportion. 
o  lo  o      lo 

In  general,  if  four  numbers  a,  5,  c,  and  d  are  in  propor- 
tion^ they  are  in  proportion  by  subtraction ;  that  is,  the  first 
antecedent  minus  its  consequent  is  to  that  consequent  as  the 
second  antecedent  minus  its  consequent  is  to  that  consequent. 

Proof.    Let  ^  =  --  (1) 

0      a 

Subtracting  1  from  each  member,  we  have 

a       ^       c      ^  a  —  h      c  —  d  ,<^. 

Here  (2)  is  said  to  be  obtained  from  (1)  by  subtraction. 


EXERCISES 

Write  by  subtraction  and  test  resulting  proportions  in  the 
numerical  exercises : 

1.  f  =  f .      3.  y^2  =  If     5.  2  :  5  =  3  :  a;.      1.  a  :b  =  m -.n. 

2.  f  =  -i/.    4.  if  =  ff     6.  7:8  =  m:7i.     8.  /:s  =  ^:L 
9.  li  a  '.  b  z=  c  :  d,  show  that  (a  —  b)  :  a  =  (c  —  d)  :  c. 


EATIO  AND  PROPORTION  289 

124.  Proportion  by  addition  and  subtraction.  If  we  write 
the  proportion  f  =  f  f  by  addition,  we  have 

¥=ff  (1) 

If  we  write  the  same  proportion  by  subtraction,  we  have 

i  =  U-  (2) 

Now  (l)-^(2)  gives  the  proportion 

J  3  _  5  2 
3     -"12- 

In  general,  if  four  numbers  a,  &,  c,  and  d  are  in  propor- 
tion^ they  are  in  proportion  hy  addition  and  subtraction;  that 
is,  the  sum  of  the  first  afitecedent  and  its  consequent  is  to 
their  difference  as  the  sum  of  the  second  antecedent  and 
its  consequent  is  to  their  difference. 

Proof.    Let  ^  =  -.  (1) 

0      a 

Then  ^-±-^  =  ^^t_^  (addition),  (2) 

0  d  ^  ^ 

and  = (subtraction).  (3) 

Dividing  (2)  by  (3),  ?^  =  |^  •  (4) 

Proportion  (4)  is  said  to  be  obtained  from  proportion  (1)  by 
addition  and  subtraction. 

Proportions  by  addition,  subtraction,  and  addition  and  subtraction 
are  often  called  proportions  hy  composition,  division,  and  composition 
and  division  respectively. 

EXERCISES 

1.  Write  Exercises  1-3,  p.  288,  by  addition  and  subtraction. 

Using  addition  and  subtraction,  solve  the  following  for  x : 
5^-^-63  a;  + V2_3- V2. 

•  5x-6~2'  •  a:-V2~3  +  V2 

2a; +  3      8  +  3  5    V^ -f  2  ^  "v^  +  3 

2a:-3~8-3*  '   ■y/x-2       V2^~3* 


290  FIRST  COUESE  IN  ALGEBEA 

6.  li  a:b  =  c:d,  show  that  3a:b  =  Sc:d. 

7.  lia:b  =  c:d,  show  that  (3  a -}- b):  b  =(Sg -\- d):  d. 

8.  If   a:b  =  c:d,  show  that  (a  +  2b)'.b  =={g  ^  2  d)\  d, 

125.  A  series  of  equal  ratios.  We  now  proceed  to  prove 
that  in  a  series  of  equal  ratios  the  sum  of  the  antecedents  is  to 
the  sum  of  the  consequents  as  any  antecedent  is  to  its  consequent; 

that  IS,  II     -  =  -  =  -,     then     =  -  =  -  =  —  . 

b     d     f  b+d+f     b     d    f 

Proof.    Setting  each  of  the  given  equal  ratios  above  equal  to  r, 

-  =  r;  -  =  r;  an(i-  =  r.  (1) 

Then  from  (1),  a  =  b-ry  (2) 

c  =  d'r,  (3) 

e=f'r.  (4) 

Adding  (2),  (3),  and  (4),      a  +  c  +  e  =  br  +  dr  +fr.  (5) 

Factoring  in  (5),  a+  c  -{■  e  =  (b  -{■  d  +  f)r.  (6) 

Therefore  l±l±l  =  r.  (7) 

b  +  d-{-f  .    ^  ^ 


Hence  by  (1),  and  (7), 


EXERCISES 

Test  the  truth  of  the  preceding  statement  in  the  following : 

1       2_4_10  2     !^_?J?_i^.  o      5_10_15 

4.  2  :(c  -\-  d)=  2 a  :(aG  -}-  ad)=  2 c  : (c"  +  cd). 

^^p(X       c       e     ,         ,-,3a-f-4:c4-5^       ^ 
b      d      f  3b-^4:d  -\'  of      b 


RATIO  AND  PROPOETION  291 

PROBLEMS 

1.  A  tract  of  land  is  purchased  by  two  men  jointly,  one 
contributing  $2500  and  the  other  $3500.  During  a  given  year 
oil  wells  on  the  tract  produced  in  royalties  $2400.  How  much 
of  this  sum  should  each  receive  ? 

2.  Three  men  bought  a  piece  of  property,  contributing 
$1000,  $1500,  and  $2000  respectively.  If  in  selling  the  prop- 
erty the  owners  gain  $900,  how  much  should  each  receive  ? 

3.  The  perimeter  of  a-  triangle  is  39.  Two  sides  are  10 
and  16.  The  other  side  is  divided  into  two  parts  which  are  in 
the  ratio  of  these  two.    Find  the  parts  of  the  third  side. 

4.  The  sides  of  a  triangle  are  15,  20,  and  25.  The  side  20 
is  divided  into  parts  which  are  proportional  to  the  other  two 
sides.    Find  these  two  parts. 

Fact  from  Geometry,  If  one  triangle  is  similar  to  another, 
the  sides  of  the  first  taken  in  any  order  are  proportional  to 
the  sides  of  the  second  taken  in  the  same  order. 

5.  The  sides  of  a  triangle  are  15,  21,  and  33  respectively. 
In  a  similar  triangle  the  side  which  corresponds  to  21  in  the 
first  triangle  is  14.  Find  the  other  two  sides.  Compare  the  ratio 
of  any  pair  of  corresponding  sides  with  that  of  the  perimeters. 

6.  The  sides  of  a  triangle  are  12,  28,  and  36.  The  perimeter 
of  a  similar  triangle  is  |  that  of  the  given  triangle.  Find  the 
sides  of  the  second  triangle. 

Facts  from   Geometry.    A  line    parallel   to   one    side   of   a 
triangle  divides  the  other  two  sides  into  four 
parts  which  are  proportional,  and  the  triangle 
cut  off  is  similar  to  the  first. 

Thus,  in  triangle  ABC,  if  DE  is  parallel  to 

side  BC,  then  D/. \e 

BD_CE  AD       AE       DE 

DA~  EA'     ^^        AB"  AC~  BC' 


292  FIEST  COUESE  IN  ALGEBEA 

7.  If  in  the  triangle  ABC  on  page  291, 

(a)  AD  =12,  DB  =  9,  and  AE  =  8,  find  EC, 

(b)  AD  =  12,  AB  =  21,  and  DE  =  8,  find  BC, 

(c)  DB=%  AD  =  12,  and  AC  =U,  find  AE. 

8.  A  flagstaff  casts  a  shadow  100  feet  long  at  the  same 
time  that  a  man  5  feet  8  inches  high  casts  a  shadow  14  feet 
and  2  inches  long.    How  high  is  the  flagstaff  ? 

9.  In  the  triangle  ABC,  having  a  right  angle  at  C,  CD  is 
perpendicular  to  AB  and  so  divides  AB  that  CD  is  a  mean 
proportional  between  ^i)  and  DB.  c 

(a)  If  AD  =  3,  and  i^i)  =12,  find  CD, 

(b)  If  ^D  =  4,  and  CD  =  6,  find  DB.      ATd' 

(c)  If  AB  =  26,  and  CD  =  12,  find  AD  and  DB. 

(d)  If  ^^  =  34,  and  CD  =  15,  find  AD  and  D5. 

10.  The   distance   AB    between    two    points    on    opposite 
banks  of  a  river  is  wanted.    Stakes  were  set  at  E,  B,  D,  and 
C  so  that  BE  was  parallel  to  CD  and  so  ^ 
that  ABC  and  ^£JD  were  straight  lines.               ^^^^Tx 
The  measured  values  of  DC,  CB,  and  BE     ^^C^  s^^^,   \ 
were    560  feet,    250  feet,    and    360  feet        ^^     ^^^^  ^ 
respectively.    What  was  the  computed  value  of  BA  ? 

11.  Two  men  start  at  the  same  time  and  travel  in  opposite 
directions  at  rates  in  the  ratio  of  14 :  16.  If  in  14  hours  they 
are  105  miles  apart,  find  the  rate  of  each. 

12.  The  diameters  of  the  earth  and  of  the  sun  are  in  the  ratio 
1:108.  Given  that  the  volumes  of  spheres  are  proportional  to 
the  cubes  of  their  respective  diameters,  find  the  ratio  of  the 
volumes  of  the  earth  and  the  sun. 

13.  Given  that  the  surfaces  of  spheres  are  proportional  to  the 
squares  of  their  respective  radii,  find  the  ratio  of  the  surfaces 
of  the  earth  and  the  sun. 


SUPPLEMENTARY  TOPICS 

126.  Highest  common  factor.  In  arithmetic  the  greatest  com- 
mon divisor  of  two  or  more  integers  is  the  greatest  integer 
which  is  exactly  contained  in  each  of  the  given  integers. 

This  greatest  common  divisor  for  two  or  more  integers,  or 
highest  common  factor  (H.C.F.),  as  the  corresponding  idea  is 
commonly  designated  in  algebra,  can  often  be  obtained  by 
inspection ;  that  is,  without  writing  the  integers  as  the  indi- 
cated product  of  their  prime  factors. 

Thus  the  H.C.F.  of  5  and  15  is  6  ;  of  30  and  42  is  6  ;  of  6  a  and  8  aP-  is  2  a. 

The  degree  of  a  polynomial  is  the  same  as  that  of  its  term 
of  highest  degree.    (See  page  75.) 

Thus  ^a^xy  +  5  ax^  —  3  a^xyz^  is  of  the  seventh  degree. 

The  highest  common  factor  (H.C.F.)  of  two  or  more  monomials 
or  polynomials  is  the  expression  of  highest  degree,  with  the 
greatest  numerical  coefficient,  which  is  an  exact  divisor  of  each. 

Thus  the  H.C.F.  of  28a263  and  42^262  is  14a262.  The  H.C.F.  of 
x^  —  4 X  and  x^  —  6 x2  +  6 x  is  x(x  —  2),  or  x2  —  2  x. 

EXAMPLES 
1.  Find  the  H.C.F.  of  12x^y^Zy  ^^  x^yH\  and  108  a;Y^*- 
Solution.  Factoring,  we  have 

72x82/^2;  =  28. 32.  xs?/^^, 
90x22/3;j4  =  2.32.6x22/V, 
108x42/V  =  22.33.xV2^ 
Here  the  highest  power  of  2  common  to  each  expression  is  the  first ; 
of  3,  the  second  ;  of  x,  the  second ;  of  y,  the  third ;  and  of  z^  the  first. 
Therefore  the  H.C.F.  of  the  three  expressions  is  2  •  3^  .  x2?/82;,  which 
equals  l^z'^yH. 

293 


294  FIRST  COUESE  IN  ALGEBRA 

2.  Find  the  H.C.F.  of  9  a:^  -  36  x'^  and  ^x'^-\2x^+  12  xK 
Solution.   Factoring,  we  have 

9x* -  36x2  =  32x2  (X  +  2)  (X  -  2), 
3x7- 12x6  + 12x5=  3x5(x- 2)2. 
Therefore  the  H.C.F.  is  3x2  (x  -  2),  which  equals  3x3  -  63.2, 

The  method  used  in  the  preceding  solutions  for  finding  the 
H.C.F.  of  two  or  more  monomials  or  polynomials  is  stated  in  the 

Rule,  Separate  each  expression  into  its  prime  factors.  Then 
find  the  pjroduct  of  such  factors  as  occur  in  each  expression, 
using  each  prime  factor  the  least  number  of  tiines  it  occurs  in 
any  one  expression. 

If  two  or  more  polynomials  have  no  common  factor  other 
than  1,  then  1  is  the  H.C.F.  of  the  polynomials,  and  they  are 
said  to  be  prime  to  each  other. 

EXERCISES 
Find  the  H.C.F.  of  the  following : 

1.  50,  75.          5.  126,  162,  198.  9.  10  a\  4  a\  12  a\ 

2.  24,  42.         6.  105,  189,  231.  10.  24^2,  18  a;^ 

3.  72,  60.          7.  208,  364,  468.  11.  IQxS  40:^2,  24x3. 

4.  90,  105.       8.  84,  210,  462,  588.  12.  60  a^x,  84  ax^,  24  aVy\ 

13.  30  a^c%  75  ac^  150  a^c%  90  a^c^x-^. 

14.  36  xy,  90  x^z,  126  x^yz,  180  xY- 

15.  72  xyH\  216  x'^yh\  504  x^^^  144  x^z^- 

16.  (a2-4),  (a2  +  4a  +  4). 

17.  (x2  -  9),  (x2  -  X  -  6). 

18.  (c^  +  c2  -  20),  (c3  +  5  c). 

19.  (Z>2  _  1),  (&8  4. 1),  (2  &2  4.  5  J  +  3). 

20.  (8-4  a),  (8  -  a%  (4  -  a''). 

21.  (3  x2  -  3  2/2),  (3  x2  -  3  xy),  (6  x^  -  6  y^). 

22.  (5  c2x2  -  20  x2),  (5  c*  +  5  c3  -  30  c%  (30  c^  -  40  c''  -  40  c^). 


SUPPLEMENTARY  TOPICS  295 


127.  The  square  of  any  trinomial.    The  ] 

[nultiplic 

a  -\-h  -\-  c 

a  -\-b  +  G 

a^  -\-     ah  -\-     ac 

^     ah               ^h^^ 

be 

■}-     ac           + 

hc  +  c" 

a^  +  2ah  +  2ac  +  h^-^2hc-{-c^ 
gives  the  formula 

(a  4.  &  _}.  c)2  =  a2  +  52  _j^  ^2  _^  2  a&  +  2  ac  +  2  6c. 
This  may  be  expressed  in  words  as  follows  : 

The  square  of  any  trinomial  is  equal  to  the  sum  of  the 
squares  of  the  terms,  plits  twice  the  algebraic  product  of  each 
term  by  each  term  that  follows  it  in  the  trinomial. 

EXERCISES 

Expand  by  the  method  indicated  above  : 

1.  (a  +  a:  +  l)^.  6.  {c  +  x  -  2)2.  11.  (a-2x-  cy. 

2.  (m^n-\-  2)2.  7.  (a  +  5  +  3)2.  12.  (3  a  +  x  +  2  cf. 

3.  (r  -  5  +  5)2.  8.  (x  +  6  +  2  a)2.  13.  (4  a  -  2  a:  +  c2)2. 

4.  {c-Vd-  3)2.  9.  (c"+  2 a  +  2)2.  14.  {^a^-\-^x-  1)2. 

5.  (a  +  c  +  xf.  10.  (a  -  a:  +  2  cf.  15.  (5  a  -  2  c2  -  3)2. 

128.  Factors  of  x^  it  y^.    By  division  we  obtain 

x^  -\-  tf* 

—  =  X*  —  x^y  -f-  x^y^  —  xy^  +  y\ 

Hence  the  sum  of  the  fifth  powers  of  two  numbers  is  exactly 
divisible  by  the  sum  of  the  numbers. 

EXERCISES 
Find  the  prime  factors  of  the  following : 

1.  a^  +  b^.  4.  x^  +  32.  7.  (c^  +  d^. 

2.  x^  +  w>.  5.  32  +  c5.  8.  mio  +  nK 

3.  a^  -f  26.  6.  243  +  a;6.  9.  r^  +  s^\ 

BE 


296  FIRST  COUESE  IN  ALGEBRA 

10.  5^5  + /^^.  13.  c5-a;5.  17.  243-a5. 

11.  x^-i/.  14.  m^-n^  18.   (x^-y. 
Hint.   Divide  hj  x  —  y.  15.  m^  —  2^  19.  a^^  —  h^ 

12.  a^-'b^  16.  /i5-32.  20.  x^^  -  y^^ 

129.  Operations  with  surds  of  different  orders.  If  the  indicated 
root  of  a  rational  number  cannot  be  taken  exactly,  the  radical 
is  sometimes  called  a  surd. 

Surds  of  different  orders  should  be  reduced  to  respectively 
equivalent  surds  of  the  same  order  before  the  operations  of 
multiplication  or  division  are  performed  with  them.  This 
process  is  also  necessary  before  the  relative  magnitudes  of 
surds  can  be  compared. 

EXAMPLES 

1.  Multiply  ^5  by  V5. 

Solution.  ^.  v^  =  4i.6i  =  4^  .5^  =  \/i^.  v^=  v^42T68  =  ^y2000^ 

2.  Divide  V2  by  v^. 

Solution.   V2  -^  \/6  =  2^  -f-  6^  =  2^  -f-  6^  =  -^23  ~-  Vq^ 

^6/23^    6l^_  6;2.3^_-yi62 
\62       \32      \    36        —3—* 

EXERCISES 
Perform  the  indicated  operations  and  simplify  results : 

1.  v^.Vs.  g        /a^    3/«  14.  V^-f-V^. 

2.  ^.V3.  *  ^^    ^^' 15.  V^^V?. 

^  ^     ,-    •  '"•  V^-Vt7-      18.  ^-.Vf. 

5.  V2o  •  VS.  ,-      „/_  , ^ 

11.  V2-^■y4.  -g      3/2^2^     ^ 

6.  -v^.^^.  12.   ^-^V2.  \  a2    •    \3* 

7.  v^5^  •  Vio^.  13.  VlO  -^  \/5.  20.  V|  -^  V|. 


SUPPLEMEKTAKY  TOPICS  297 

Arrange  the  following  in  order  of  magnitude : 

21.  </6,  V7. 

Hint.  Reduce  to  respectively  equivalent  surds  of  the  sixth  order  and 
compare  radicands. 

22.  V5,  vTl.  24.  ^9,  VE.  26.  Vl6,  ^4. 

23.  <^,  V3.  25.  VT2,  -v/io.  27.  v^,  \^,  v^. 

130.  Solution  of  the  quadratic  equation  by  formula.    If  the 

general  quadratic  equation  ax^  -{-  bx  -\-  c  =  0  is  solved  by  the 
method  of  completing  the  square,  the  result  is 

This  expression  is  a  general  result  and  may  be  used  as  a 
formula  for  the  solution  of  any  quadratic  equation. 

If  the  numbers  a,  b,  and  c  are  such  that  ^^  —  4  ac  is  negative, 
then  the  formula  would  contain  the  square  root  of  a  negative 
number,  which  is  a  kind  of  number  not  considered  in  this 
text.  In  the  exercises  that  follow,  it  will  be  assumed  that 
only  such  numerical  values  of  the  letters  involved  are  con- 
sidered as  will  not  make  P  —  4:ac  negative. 

EXERCISES 

Solve  by  formula,  and  check. 

1.  4^2+  8x  =  3. 

Solution.   Writing  in  standard  form,       4x2  +  8x  —  3  =  0. 

Comparing  v^rith  ax^  +  6x  +  c  =  0,  evidently  4  corresponds  to  a,  8  to  b, 
and  —  3  to  c.   Substituting  these  values  in  (F)  gives 


-  8  ±  V64  -  4  . 

4. (-3) 

2-4 

__  8  db  V64  +  48 

-8±Vll2 

8 

8 

= 

-8±4V7_ 
8 

l±lV7. 

Check  as  usual. 


298  EIEST  COUESE  IN  ALGEBEA 

8.  x^ -  =  0. 

Z.  x^-x-l=0.  4  4 

4.  2  0:2  +  5  a:  +  2  =  0.  9,  6x  +  7=x^. 

5.  nx^+6x  =  4:.  10.  21:r  =  l-72x2. 

6.  2x  +  4:  =  x^  11.  2  0:2  +  3  :r  -  1  =  0. 

7.  0:2  +  a:  =  1.  12.  5o:=  3.1:2  +  1. 

13.  o:2+o:V5  =10. 
14.   2  y^2^2  :^  y^.^  +  1, 

Solution.    Writing  in  standard  form,  2  A:2x2  —  Jcx  —  1  =  0. 

Then  a  =  2k'^,b  =  —  k,  and  c  =  -l. 
Substituting  tliese  values  in  the  formula  (F), 

_-{-  lc)±  V(-  A:)2  -  4  .  2  A;2  (-  1) 
^~  2.2A;2 


fc±Vfc2  4-8^:2      A:±3fc      1       ^        1 

X  = = =  -  and 

4k^  4:k^         k  2k 

Check  as  usual. 

15.  o:2+2A-o:-3^2:^0.  20.  aV- S  ahx  +  2b^  =  0. 

16.  x^-2xV^-da  =  0.  21.  p^x^  -  i  pqx  -  12  q^  =  0. 

17.  2a^=9ax  -^  6x^.  22.  2  w2o;2  +  5  nmx  +  8  m2  =  0. 

18.  3  0:2  +  &o:  -  10  Z^2  ^  0.  23.  6  ^2^2  +  19  hkx  =  7  B. 

19.  ex  =  6o:2-  c2.  24.  3  aV-  6  abx  +  2b^=  0. 
25.  0:2+  2o:  =  Ao:  +  2^. 

Hint,   x^  +  (2  -  h)x  -  2h  =  0.    Then  a  =  1,  6  =  2  -  /i,  and  c  =  -  27i. 
Substituting  these  values  in  (F), 


X  =  -(2-A)±V(2-fe)'-4.1.(-2fe)    ^^_ 

26.  0:2  +  (p  —  q)  X  —  pq  =  0.  29.  mno:2  +  4  no:  —  3  mx  =  12. 

27.  2o:2+a&  =  2  5o:  + ao:.  30.  x'^-2hx  =  g-h\ 

28.  r5o:2  +  50:  =  3  ro:  +  3.  31.  ^o:2  +  A:o:  -  2  =  2  0:. 


SUPPLEMENTARY  EXERCISES  INVOLVING 
EQUATIONS 

Solve: 

l.^^-^  =  2.  6.^  +  5  =  1. 

5  3  XX-. 

x-\-2  x-2^\                    ^    ?_^_i. 

9  4          6                     '  X      X      ^ 

^    2x-2>  1-x  ^         ^    4       3        13 

3  4                                  oj      4a?       8 

7  5  35     '  2a5"^9      3x* 

^-  -3-  +  ^-- ^-  =  15-     ^^•-V--4  +  -20--^- 

11.   1(^+1) +J(a._37^)  =  0. 

4a; -3      37  /4ar  +  15\ 
6  4  ~  18  V      14      / 

13.  ^(2x2-5;:c  +  3)=f(x2-2aj  +  l). 

3a;-4     a;  +  5     1  x-\  x -{- 7         x  —  5^1S 

2      ~     3     ""2*     4     '  2x-4.      2a;  +  l       4' 

g;  —  4  _      ag  a?  +  3      x  —  4  _  2  a?  —  5 

X      ~  a;  +  20*  cc  +  1  "^  a;  +  5  ~   a;  -  3  ' 

3a;-4_6a;-3  3a; -4      2a; -1  7- 

a;  +  l    ~2a;  +  6'  2a;H-3      4a;4-6~      6* 

17.  SJzi^^J^.  22.  ^-:^-^±^  +  ^  =  o. 


a;H-la;H-9  '  x -{- 2      x  —  5       3 

5  —  a;.  05  —  1      2  o«^  +  3.a;  —  1 

x^3^x-\-l      3  'a;-3      a;  +  4 


299 


800  FIRST  COURSE  IN  ALGEBRA 

X  X  —  1      8a?  —  37 

x2-9~aj  +  3~8(3-x) 

3(0? +  2) 1      _6(x-S) 

ar^-4         2-ir         2 -{- x 

^  +  o"  ^  o" 


26.  — ^-:^  =  x. 
-4aj  +  3      5 


27. 

28. 


6ic  +  8      8 

0^  1 


3x  —  4      4  —  0?      1  +  a; 

29.  Ax-  7.1  =  1.3 aj  +  .1.  31.  .14  -.9x  =  .15 x. 

30.  .S(l.lx-5)  =  .03x-.6,      32.  .4  (x  -  3)  =  .5  (x  +  1). 

33.  .2(4.1x-.7)  =  .7(.lx-h.8). 

34.  3.15  X  -  4.6  +  a;  -  3.2  =  .05  x  +  .4. 

35.  .173  X  -  4.68  -  .13  x  +  2.561  =  .32  x  +  .897  -  .335  x 

36.  .65  (.4  X -.7)  =  .25  (.9-. 32  x). 

Ax -7      x-5.6         ^^    .3x~.9      4.4-6.3x 
38. = 


40. 
41. 
42. 


Oi» 

6 

5 

39. 

.175x-f-.21 

.8 

2.1 

-.8x 

.02X  +  .08 

.4x 

+  .9 

~.09  4-.01x 

X  — 

a       X 

-b 

3.2 


+  ^(.9x-.16)  =  .908x. 


a  _x 
X      a 


X        ft^ 


44. 


X  —  a  _a  —  b 
X  —  b       a  -\-b 


45. 

X           a  -f  ^ 

46. 

X 

ax  =  -• 

47. 

X  —  a       X  -{-  b 
a              b 

48. 

49. 

X         X 

i^ 


ax  +  fc<: 


EXERCISES  INVOLVING  EQUATIONS         301 

ir^        -^      .      ^  O  «.«        ^  ~   ^  X  -{-2'b  _ 

50.  -  4-  -  =  3.  52.  — ; —  = 3. 


51.  -^--F-  =T^-  53.  T  = 


X 

b>_ 

X 

=  3. 

a 

a 

1 

3^ 

~5 

X 

15 

54. 

X 

1 

+ 

—  a 

1 
x-b 

55. 

X 

17a 
12  c 

1      ^' 

'  2(^x 

h      c  —  X 

(x  —  d)(x  —  b) 
=  0. 

56.  A  bubble  of  air  of  volume  v  units  rising  from  a  depth 
of  d  feet  below  the  surface  of  the  water  expands  to  the 
volume  V  units  at   the  top  of  the  water  according   to   the 

formula  V  =  — r-r —  v.    Express  v  in  terms  of  d  and  V,    If  a 

bubble  starts  from  100  feet  below  the  surface  and  has  a  volume 
of  .1  of  a  cubic  inch  at  the  top,  find  its  volume  when  it  started. 

57.  The  formula  V=—^ Ogives  the  volume  of  a 

solid  whose  height,  upper  base,  lower  base,  and  mid-section 
are  A,  b,  B,  and  m  respectively.  Eind  m  in  terms  of  the 
other  letters. 

58.  Given  the  formula  5  =  ^^  +  16^  for  the  distance  (5) 
covered  by  a  body  projected  downward  with  a  velocity  v 
in  a  time  t,  the  units  being  feet  and  seconds,  express  v  in 
terms  of  s  and  t. 

59.  Given  T — ;  =  0.  Express  c?  in  terms  of  the  other  letters. 

b      d 

60.  Given  — ; —  =  — - —  Express  d  in  terms  of  the  other 
1  i...  b  d  "^ 

letters.  a      b 

61.  If  a  =  2 ??  and  ^  =  3 1,  find  the  value  of  ^ ^. 

62.  If  a  =  ^,  ^  =  2  ^,  c  =  3  ^,  find  in  terms  of  t : 


(a-b){c-b)   '   (b-a){c-a)       (a-c)(b-c) 


302  FIRST  COURSE  IN  ALGEBRA 

63.  Separate  60  into  two  parts  such  that  ^  the  greater  minus 
^  the  less  equals  22. 

64.  Separate  91  into  two  parts  such  that  their  quotient 
is  5i 

65.  Separate  70  into  two  parts  such  that  40  exceeds  -|  of 
the  one  by  as  much  as  the  other  exceeds  20. 

66.  The  sum  of  two  numbers  is  13.  Two  thirds  the  greater 
plus  I  the  less  equals  9.   What  are  the  numbers  ? 

67.  Two  thirds  a  man's  age  now  equals  f  his  age  25  years 
ago.   What  is  his  age  ? 

68.  The  square  of  a  certain  number  is  17  greater  than  f 
of  the  product  of  the  next  two  consecutive  numbers.  Find 
the  number. 

69.  The  length  of  a  certain  rectangle  is  2^  times  the  width. 
If  it  were  a  yard  shorter  and  a  yard  and  a  half  wider,  its  area 
would  be  234  square  feet  greater.  Find  the  dimensions  of  the 
rectangle. 

70.  A  square  court  has  the  same  area  as  a  rectangular  court 
whose  length  is  8  yards  greater  and  whose  width  is  5  yards 
less.    Find  the  dimensions  of  the  square  court. 

71.  A  can  do  a  piece  of  work  in  2  days  ;  B,  in  2^  days  ;  C,  in 
3^  days.   How  long  will  it  take  them,  working  together  ? 

72.  The  difference  between  |^  of  a  certain  number  and  J  of 
it  is  20.    Find  the  number. 

73.  Four  thousand  dollars  of  Mr.  A's  income  is  not  taxed. 
All  of  his  income  over  that  amount  is  taxed  2%,  and  all 
above  ten  thousand  dollars  is  taxed  2%  in  addition.  He  pays 
a  tax  of  |200.    What  is  his  income  ? 

Sx^  -\-  X 2 

74.  Given  t  = >  substitute  this  value  for  t  in 

3^  +  90^  +  2       .    .  ^.7/ 
-^-^3^^;^  and  simplify. 


EXERCISES  mVOLVING  EQUATIONS         303 

75.  If  -  -f-  7  =1,  and  if  -  =  c,  find  b  in  terms  of  a  alone, 

abb  ' 

and  a  in  terms  of  c  alone. 

76.  Working  8  hours  per  day,  A  can  do  a  certain  piece  of 
work  in  7  days  4  hours,  B  in  3  days  6  hours.  After  a  third 
man  had  done  a  quarter  of  the  work,  he  struck,  and  A  and  B, 
taking  his  place,  finished  the  work  together.  How  long  did 
it  take  them? 

77.  A  and  B  start  at  the  same  time  from  two  towns  100 
miles  apart  and  travel  toward  each  other.  Their  respective 
rates  are  8  and  12  miles  per  hour.  Before  they  meet,  A  rests 
2^  hours  and  B  rests  5  hours.  How  far  does  each  go  and  how 
long  is  it  before  they  meet  ? 

78.  A  and  B  leave  the  same  place  at  the  same  time  for  a 
point  90  miles  distant,  A  traveling  twice  as  fast  as  B.  Upon 
reaching  their  destination  A  turns  back  and  meets  B  6  hours 
from  the  start.   What  are  their  rates  ? 

79.  A  man  rows  upstream  and  back,  a  total  distance  of 
30  miles,  in  9  hours.  His  rate  upstream  was  half  his  rate 
downstream.  Find  the  rate  of  the  current  and  his  rate  in 
still  water. 

80.  A  bolt  of  cloth  is  ^Dought  f or  $324.  Eight  yards  are  cut 
off  for  use  as  samples,  and  the  remainder  sold  at  an. advance 
of  $1  per  yard,  yielding  a  profit  of  |76.    Find  the  cost  per  yard. 

81*.  Two  autdmobiles  each  travel  72  miles,  one  going  4  miles 
per  hour  faster  than  the  other  and  making  the  run  in  15  minutes 
less  time.    Find  the  rate  of  each. 

82.  A  and  B  start  at  the  same  time  and  place,  but  travel 
in  opposite  directions.  B  is  delayed  2  hours  on  the  way  and 
travels  1  mile  per  hour  slower  than  A.  At  the  end  of  a  certain 
time  they  are  172  miles  apart.  If  A  has  then  traveled  28  miles 
farther  than  B,  find  the  number  of  hours  since  they  started. 


304  FIEST  COUESE  IN  ALGEBRA 

83.  A  and  B  together  can  do  a  piece  of  work  in  3f  days. 
B  alone  can  do  it  in  3  days  less  than  A.  Find  the  number  of 
days  required  by  each. 

84.  At  what  time  between  3  and  4  o'clock  are  the  hands  of 
the  clock  together  ? 

Hint.  Let  x  =  the  number  of  minutes  after  3.  Then  the  hour 
hand  covers  a:  — 15  spaces  and  the  minute  hand  x  spaces,  while  the 
latter  goes  12  times  as  fast  as  the  former. 

85.  At  what  time  between  7  and  8  are  the  hands  of  the 
clock  together? 

86.  At  what  time  between  9  and  10  are  the  hands  of  the 
clock  together  ? 

87.  At  what  time  between  1  and  2  are  the  hands  of  the 
clock  in  the  same  straight  line  but  in  opposite  directions  ? 

88.  How  long  will  it  take  to  go  m  miles  at  d  miles  per  hour  ? 

89.  What  is  the  surface  of  a  cubical  box  whose  edge  is 
d  inches  ? 

90.  Express  in  dollars  ^%  of  x  dollars. 

91.  If  the  average  height  of  k  boys  is  n  inches,  what  is  the 
sum  of  their  heights  in  yards  ? 

92.  If  2/1  + 1  bolts  weigh  x  pounds,  what  is  the  weight  of 
n  of  them  ? 

93.  To  cook  cereal  in  a  fireless  cooker,  one  uses  n-\'2  cups 
of  water  for  n  cups  of  dry  cereal.  How  many  cups  of  cereal 
may  be  used  in  a  dish  that  holds  k  cups  in  all  ? 

94.  If  it  takes  a  man  h  hours  to  do  a  piece  of  work,  what 
portion  of  the  work  can  he  do  in  1  hour  ?  What  portion  of 
the  work  would  n  men  do  in  1  hour  ?  What  portion  would  k 
men  do  in  t  hours  ? 

95.  If  it  takes  m  men  h  hours  to  do  a  piece  of  work,  how 
long  will  it  take  r  men  to  do  it  ? 


EXEECISES  INVOLVING  EQUATIONS         305 

96.  A  man  buys  bananas  at  c  cents  a  dozen  and  sells  them 
for  k  cents  each.   What  does  he  gain  on  d  dozen  ? 

97.  A  man  buys  oranges  for  d  dollars  per  hundred  and  sells 
them  h  for  a  quarter.  *  How  much  does  he  gain  on  h  hundred  ? 

98.  A  man  bought  x  articles  for  c  cents  per  hundred.    He 
sold  them  all  for  $6.    How  many  dollars  did  he  lose  ? 

99.  A  man  buys  goods  for  x  dollars  and  sells  them  for 
X  —  y  dollars.    What  is  his  per  cent  of  loss  ? 

100.  If  y  yards  of  ribbon  cost  d  cents,  find  the  cost  of 
X  yards. 

101.  If  y  yards  of  ribbon  cost  d  cents,  how  many  yards  can 
be  bought  for  k  dollars  ? 

102.  A  man  buys  goods  for  d  dollars  and  sells  them  for 
h  dollars.    What  is  his  per  cent  of  gain  ? 

103.  One  man  can  do  a  piece  of  work  in  d  days,  another 
can  do  the  same  work  in  /  days.  How  many  days  will  it  take 
both,  working  together  ? 

104.  If  it  takes  h  hours  to  mow  a  acres,  how  many  days  of 
10  hours  each  will  it  take  to  mow  h  acres  ? 

105.  A  train  goes  /  feet  in  t  seconds.  If  this  equals  m  miles 
per  hour,  write  an  equation  involving  /,  t,  and  m. 

106.  If  n  men  can  do  a  piece  of  work  in  d  days,  how  many 
men  would  it  be  necessary  to  hire  if  the  work  had  to  be  done 
in  k  days  ? 

107.  A  transport  plying  betweeil  two  ports  is  under  fire  for 
y  yards  of  the  way.  If  she  steams  k  knots  per  hour,  for  how 
many  minutes  is  she  under  fire  ?    (1  knot  =  6080  feet.) 


INDEX 


Abacus,  186 

Absolute  value^  2^ 

Aggregation,  signs  of,  64 

Al-jebr,  57 

Antecedent,  287 

Arabic  notation,  1,  9,  186 

Arabs,  5, 11,  67, 93, 186,  249, 254, 261 

Archimedes,  274 

Arrangement,  75 

Associative  Law  of  Multiplication, 

70,  71 
Axiom,  39,  40,  175 

Babylonians,  157 

Bacon,  Roger,  186 

Binomial,  35,  82  ;  square  of,  105 

Binomials,  product  of,  107, 109, 110 

Braces,  64 

Brackets,  64 

Cancellation,  149,  167 

Cardan,  279 

Check,  36,  42,  45,  46,  51 

Circle,  area,  15;  circumference,  191 

Coefficient,  10 

Coefficients,  literal,  95;  polynomial, 
38,96 

Commutajtive  Law  of  Multiplica- 
tion, 70 

Consequent,  287 

Constant  term,  137 

Coordinates,  203 

Cube  root,  11,  114,  251 

Cubes,  sum  or  difference  of,  133 

Cubic  equation,  141,  275 

Cylinder,  surface,  191 ;  volume,  16 

Decimal  point,  186,  249 

Decimals,  249 ;  equations  contain- 
ing, 184 

Degree,  75 ;  of  an  equation,  137, 
141 ;  of  a  polynomial,  293  ;  of  a 
term,  75,  76 

Denominator,  lowest  common,  154 


Diagonal  of  rectangle,  248 
Diophantos,  278,  279 

Egyptians,  157 

Equation,  6,  39,  54,  95 ;  graph  of, 
205  ;  indeterminate,  217  ;  in  one 
unknown,  55 ;  in  several  un- 
knowns, 217  ;  in  two  variables, 
202  ;  of  condition,  54  ;  of  fourth 
degree,  275  ;  of  second  degree, 
137 ;  of  third  degree,  141 ;  root 
of,  55,  218,  264 ;  solution  of,  41 ; 
solving,  39 ;  translation  of  prob- 
lem into,  60,  82.  See  also  Linear, 
Quadratic,  Cubic 

Equations,  containing  fractions, 
175,  180;  containing  decimals, 
184  ;  incompatible,  221 ;  indeter- 
minate system  of,  221 ;  literal, 
95, 190 ;  literal,  in  two  unknowns, 
231 ;  radical,  264.  See  also  Sys- 
tems, Simultaneous  systems 

Euclid,  32,  157,  261 

Exponent,  9,  210 

Exponents,  fractional,  251 ;  law  of, 
in  multiplication,  71 ;  law  of,  in 
division,  87 

Extremes,  283 

Factor,  9,  10  ;  common  monomial, 

.  116;.highestcomnion,  293;  ration- 
alizing, 266;  zero,  138 

Factoring,  113  ff.,  295  ;  solution  of 
equations  by,  137  ff. 

Falling  body,  17 

Fractions,  1 ;  addition  of,  157 ;  al- 
gebraic, 148  ;  changes  of  sign  in, 
162,  163  ;  complex,  172  ;  division 
of,  170;  equations  containing, 
175,  180 ;  equivalent,  154 ;  his- 
tory of,  157 ;  lowest  common  de- 
nominator of,  154  ;  lowest  terms 
of,  149 ;  multiplication  of,  167 ; 
operations  with,  167  j  reduction 


307 


308 


FIRST  COUESE  IN  ALGEBRA 


of  mixed  expression  to,  165 ; 
simultaneous  systems  containing, 
224 ;  subtraction  of,  157 ;  terms 
of,  148 

Germans,  5 

Graph,  200 ;  of  an  equation,  205 ; 

of  linear  equation,  205,  206,  208 
Graphical  representation,  of  linear 

system  in  three  unknowns,  239 ; 

of  statistics,  210 
Greeks,  93,  157 

Harriot,  5 

Highest  common  factor,  293 

Hindus,  5,  21,  28,  29, 157,  186,  261, 

279 
Hypotenuse,  247 

Identity,  54  ;  sign  of,  55 

Index,  11,  251 

Integer,  1,  250;  consecutive,  44; 
consecutive  even,  44;  consecu- 
tive odd,  44;  even,  44;  odd,  44; 
positive,  27 

Interest,  15,  186;  simple,  187 

Irrational  number,  250 

Italians,  5 

Kronecker,  Leopold,  27 

Lever,  193 

Linear  and  quadratic  equations, 
systems  of,  279 

Linear  equation,  137;  in  two  vari- 
ables, 208,  217,  224 

Linear  systems  in  three  unknowns, 
236 

Literal  coefficients,  95;  equations 
with,  95,  190,  231,  277 

Lowest  common  denominator,  154 

Lowest  common  multiple,  152 

Lowest  terms,  149 

Means,  283 

Member,  left,  39;  right,  39 

Monomial,  33 ;  cube  root  of,  114 ; 
square  root  of,  113 

Monomial  denominator  in  equa- 
tions, 175 

Monomial  factor,  116 


Monomials,  addition  of,  33,  34 ; 
division  of,  87 ;  division  of  poly- 
nomials by,  89 ;  multiplication 
of,  72  ;  multiplication  of  poly- 
nomials by,  73 ;  subtraction  of,  49 

Moors,  186 

Morse,  S.  F.  B.,  275 

Motion,  uniform,  97,  192 

Multiple,  lowest  common,  152 

Napier,  Sir  John,  186 

Negative  numbers,  19,  20,  210,  279. 

See  also  Positive  and  negative 

numbers 
Number,  imaginary,  250 ;  irrational, 

250 ;    rational,    250 ;    unknown, 

6,  39 
Numerals,  1,  9  ;  arable,  9 
Numerical  value,  15,  22 

Operation,  signs  of,  1,  5,  30 
Operations,  order  of,  13 
Order,  251 ;  in  addition,  34 
Ordinate,  202 
Origin,  203 
Oughtred,  5 

Parentheses,  10,  13,  30,  34,  64,  79, 
82  ;  insertion,  68  ;  removal,  64 

Percentage,  186 

Points,  plotting,  204 

Polynomial,  35  ;    degree  of,  293  ; 

Polynomial  denominators  in  equa- 
tions, 180 

Polynomials,  addition  of,  35  ;  divi- 
sion of,  by  monomials,  89  ;  divi- 
sion of,  91 ;  multiplication  of,  by 
monomials,  73 ;  multiplication  of, 
74 ;  prime  to  each  other,  152 ; 
subtraction  of,  50  ;  with  common 
monomial  factor,  116 

Positive  and  negative  numbers,  20 ; 
addition  of,  21  ;  division  of,  28  ; 
multiplication  of,  26 ;  subtrac- 
tion of,  23 

Power,  75 

Powers,  ascending,  75 ;  descend- 
ing, 75 

Prime,  113,  152,  293 

Product,  9,  10,  70,  71,  93,  107,  109, 
110  ;  of  sum  and  difference,  107 


INDEX 


309 


Proportion,  283  ff. 
Proportional,  fourth,  284;   mean, 
285 ;  third,  286 

Quadratic  and  linear  equations, 
systems  of,  279 

Quadratic  equation,  137,  275  ;  con- 
stant term  of,  137;  history  of, 
278 ;  in  two  unknowns,  279 ; 
pure,  272  ;  solution  of,  by  com- 
pleting the  square,  270;  solution 
of,  by  factoring,  137;  solution 
of,  by  formula,  297  ;  with  literal 
coefficients,  277 

Kadical,  250 ;  sign,  11 

Radical  equations,  264 

Radicals,  addition  of,  259;  conju- 
gate, 266  ;  division  of,  266  ;  mul- 
tiplication of,  261 ;  simplification 
of,  253  ;  subtraction  of,  259 

Radicand,  251 

Radicands,  fractional,  256 

Raleigh,  Sir  Walter,  5 

Ratio,  282 

Rational  number,  250 

Rationalizing  factor,  266 

Recorde,  5,  242 

Roman  notation,  186 

Romans,  157 

Root,  cube,  11,  114;  of  an  equa- 
tion, 55,  217,  264 ;  principal,  251 ; 
principal  square,  114;  square, 
11,  113 

Roots,  cube,  251 ;  even,  114;  fourth, 
114;  of  a  monomial,  113,114,115; 
rejected,  143  ;  set  of,  218  ;  sixth, 
114 

Set  of  roots,  218 

Sign,  changes  of,  in  fraction,  162, 
163 ;  double,  114 ;  of  equality, 
2,  5 ;  of  identity,  55  ;  radical,  11 

Simultaneous  systems,  218;  con- 
taining fractions,  224 ;  solution 
of,  by  addition  and  subtraction, 
219 ;  solution  of,  by  substitution, 
222 

Solution,  definition  of,  218 ;  of 
equations,  15,  41  ;  of  equations 
by  factoring,  137  ff.;  graphical, 


of  linear  equations  in  two  varia- 
bles, 208  ;  of  problems,  6,  45  ;  of 
systems,  219,  222 

Solutions,  rejected,  143 

Solving  an  equation,  39 

Square,  of  binomial,  105;  of  tri- 
nomial, 295;  perfect,  121;  tri- 
nomial, 121 

Square  root,  11 ;  historical  note  on, 
249;  of  algebraic  expressions, 
240 ;  of  numbers,  243 ;  principal, 
114 

Squares,  difference  of,  124 

Stifel,  11,  279 

Subscripts,  191 

Substitution,  solution  of  systems 
by,  222 

Sum,  algebraic,  23 

Surd,  296 

System,  indeterminate,  221 ;  in 
three  unknowns,  236 ;  of  equa- 
tions, 217;  simultaneous,  218 

Systems,  containing  fractions,  224  ; 
of  linear  and  quadratic  equa- 
tions, 279 

Term,  33 

Terms,  dissimilar,  34 ;  of  fraction, 

148 ;  similar,  33 
Theon,  249 
Transposition,  56 
Trapezoid,  146;   altitude  of,  146; 

area  of,  146 
Triangle,  altitude  of,  145 ;  area  of, 

15,   145,  191 ;    equilateral,   248, 

258 ;  isosceles  right,  258 ;  right, 

247,  258,  259 
Triangles,  similar,  291 
Trinomial,  35 ;  general  quadratic, 

130 ;  quadratic,  127 ;  square  of 

a,  295 
Trinomial  squares,  121 

Unknown,  6,  39,  95 

Variable,  217 
Vinculum,  64 

Zero,  as  denominator,  161 ;  as 
factor,  138  ;  division  by,  29,  40 ; 
multiplication  by,  29 


ANNOUNCEMENTS 


TEXTBOOKS   IN   MATHEMATICS 

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Baker:  Elements  of  Solid  Geometry 

Barker  :  Computing  Tables  and  Mathematical  Formulas 

Beman  and  Smith  :   Higher  Arithmetic 

Betz  and  Webb  :   Plane  Geometry 

Betz  and  Webb :  Plane  and  Solid  Geometry 

Betz  and  Webb  :   Solid  Geometry 

Breckenridge,  Mersereau,  and  Moore :  Shop  Problems  in 

Mathematics 
Cobb :  Elements  of  Applied  Mathematics 
Hawkes,  Luby,  and  Touton :  Complete  School  Algebra 
Hawkes,  Luby,  and  Touton  :   First  Course  in  Algebra  (Rev.) 
Hawkes,  Luby,  and  Touton :  Second  Course  in  Algebra 

(Revised  Edition) 
Hawkes,  Luby,  and  Touton :  Plane  Geometry 
Moore  and  Miner :  Practical  Business  Arithmetic  (Revised 

Edition) 
Moore  and  Miner  :  Concise  Business  Arithmetic 
Morrison :  Geometry  Notebook 

Powers  and  Loker :  Practical  Exercises  in  Rapid  Calculation 
Robbins :  Algebra  Reviews 
Schorling  and  Reeve  :  General  Mathematics 
Smith :  Algebra  for  Beginners 
Wentworth :  Advanced  Arithmetic 
Wentworth :  New  School  Algebra 
Wentworth  and  Hill :  First  Steps  in  Geometry 
Wentworth- Smith  Mathematical  Series 

Junior  High  School  Mathematics 

Higher  Arithmetic 

Academic  Algebra 

School  Algebra,  Books  I  and  II 

Vocational  Algebra 

Commercial  Algebra 

Plane  and  Solid  Geometry.    Also  in  separate  editions 

Plane  Trigonometry 

Plane  and  Spherical  Trigonometry 


GINN  AND  COMPANY  Publishers 


HAWKES,  LUBY,  AND  TOUTON'S 
ALGEBRAS 

By  Herbert  E.  Hawkes,  Professor  of  Mathematics  in  Columbia  University, 

William  A.  Luby,  Head  of  the  Department  of  Mathematics,  Junior 

College  of  Kansas  City,  and  Frank  C.  Touton,  State 

Supervisor  of  High  Schools,  Madison,  Wis. 

FIRST     COURSE    IN    ALGEBRA    (Revised  Edition)     i2mo,  cloth, 
302  pages,  illustrated 

SECOND    COURSE   IN   ALGEBRA  (Revised  Edition)  i2mo,  cloth, 
viii  +  277  pages,  illustrated 

COMPLETE    SCHOOL    ALGEBRA  (Revised  Edition)  i2mo,  doth, 

ix  +  507  pages,  illustrated 

THE  Hawkes,  Luby,  and  Touton  Algebras  offer  a  fresh  treat- 
ment of  the  subject,  combining  the  best  in  the  old  methods  of 
teaching  algebra  with  what  is  most  valuable  in  recent  developments. 
The  authors*  unhackneyed  and  vital  manner  of  presenting  the  sub- 
ject makes  a  sure  appeal  to  the  interest  of  the  student,  while  their 
genuine  respect  for  mathematical  thoroughness  and  accuracy  gives 
the  teacher  confidence  in  their  work. 

Among  the  distinctive  features  of  these  algebras  are  the  correla- 
tion of  algebra  with  arithmetic,  geometry,  and  physics;  the  liberal 
use  of  illustrative  material,  such  as  brief  biographical  sketches  of  the 
mathematicians  who  have  contributed  materially  to  the  science ;  early 
and  extended  work  with  graphs ;  and  the  introduction  of  numerous 
"  thinkable  "  problems.  Prominence  is  given  the  equation  through- 
out, and  the  habit  of  checking  results  is  constantly  encouraged. 
Thoroughness  is  assured  by  frequent  short  reviews. 

The  aim  has  been  to  treat  in  a  clear,  practical,  and  attractive  man- 
ner those  topics  selected  as  necessary  for  the  best  secondary  schools. 
The  authors  have  sought  to  prepare  a  text  that  will  lead  the  student 
to  think  clearly  as  well  as  to  acquire  the  necessary  facility  on  the 
technical  side  of  algebra.  The  books  offer  a  course  readily  adapt- 
able to  the  varying  conditions  in  different  schools  —  the  "  Complete 
School  Algebra  "  comprising  a  one-book  course  with  material  suffi- 
cient for  at  least  one  and  one-half  year's  work,  and  the  "  First  Course" 
and  "Second  Course"  providing  the  same  material,  but  slightly 
expanded,  in  a  two-book  course. 


GINN  AND   COMPANY  Publishers 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  staipped  below. 


LD  21-95m-ll,'50(2877sl6)476 


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